Symmetry, Equilibria, and Robustness in Common-Payoff
Games
Scott Emmons
UC Berkeley
Berkeley, CA
emmons@berkeley.eduCaspar Oesterheld
Duke University
Durham, NC
caspar.oesterheld@duke.eduAndrew Critch
UC Berkeley
Berkeley, CA
critch@eecs.berkeley.edu
Vince Conitzer
Duke University
Durham, NC
conitzer@cs.duke.eduStuart Russell
UC Berkeley
Berkeley, CA
russell@berkeley.edu
ABSTRACT
Although it has been known since the 1970s that a globally optimal
strategy profile in a common-payoff game is a Nash equilibrium,
global optimality is a strict requirement that limits the result’s
applicability. In this work, we show that any locally optimal sym-
metric strategy profile is also a (global) Nash equilibrium. Applied
to machine learning, our result provides a global guarantee for any
gradient method that finds a local optimum in symmetric strategy
space. Furthermore, we show that this result is robust to pertur-
bations to the common payoff and to the local optimum. While
these results indicate stability to unilateral deviation, we neverthe-
less identify broad classes of games where mixed local optima are
unstable under joint, asymmetric deviations. We analyze the preva-
lence of instability by running learning algorithms in a suite of
symmetric games, and we conclude with results on the complexity
of computing game symmetries.
KEYWORDS
common-payoff games, coordination, collaboration, Nash equilibria,
symmetry, symmetric games, robustness
ACM Reference Format:
Scott Emmons, Caspar Oesterheld, Andrew Critch, Vince Conitzer, and Stu-
art Russell. 2021. Symmetry, Equilibria, and Robustness in Common-Payoff
Games. In Appears at the 3rd Games, Agents, and Incentives Workshop (GAIW
2021). Held as part of the Workshops at the 20th International Conference
on Autonomous Agents and Multiagent Systems., London, UK, May 2021 ,
IFAAMAS, 17 pages.
1 INTRODUCTION
We consider common-payoff games (also known as identical inter-
estgames [ 38]), in which the payoff to all players is always the
same.1Such games model a wide range of situations involving
1This condition is relaxed in (weighted) potential games where the players’ payoffs
need only imply the same ordering of outcomes [ 33]; (weighted) potential games are
best-response equivalent to common-payoff games [15, 26].
Permission to make digital or hard copies of part or all of this work for personal or
classroom use is granted without fee provided that copies are not made or distributed
for profit or commercial advantage and that copies bear this notice and the full citation
on the first page. Copyrights for third-party components of this work must be honored.
For all other uses, contact the owner/author(s).
Appears at the 3rd Games, Agents, and Incentives Workshop (GAIW 2021). Held as part of
the Workshops at the 20th International Conference on Autonomous Agents and Multiagent
Systems., Aziz, Ceppi, Dickerson, Hosseini, Lev, Mattei, McElfresh, Zick (chairs), May
2021, London, UK .©2021 Copyright held by the owner/author(s).cooperative action towards a common goal. Under the heading of
team theory , they form an important branch of economics [ 19,20].
In AI, the common-payoff assumption holds in Dec-POMDPs [25],
where multiple agents operate independently according to policies
designed centrally to achieve a common objective. Many applica-
tions of multiagent reinforcement learning also assume a common
payoff [ 7,8,12]. Finally, in assistance games [31] (also known as
cooperative inverse reinforcement learning or CIRL games [ 13]),
which include at least one human and one or more “robots,” it is
assumed that the robots’ payoffs are exactly the human’s payoff,
even if the robots do not know what it is.
Common-payoff games lead naturally to considerations of sym-
metry in game structure—for example, the assumption that two
players’ actions produce the same effect on the common payoff.
Indeed, von Neumann and Morgenstern [ 41] and Nash [ 23] intro-
duced fairly general group-theoretic notions of symmetry, which
we adopt and explain in Section 2. More recent work has analyzed
narrower notions of symmetry [ 22,30,39]. For example, Daskalakis
and Papadimitriou [ 5] study “anonymous games” and show that
anonymity substantially reduces the complexity of finding solutions.
Finally, Ham [ 14] generalizes the player-based notion of symmetry
to include further symmetries revealed by renamings of actions.
We conjecture our results extend to this more general case, at some
cost in notational complexity, but we leave this to future work.
In games exhibiting symmetry, it is then reasonable to consider
symmetry in players’ strategies. (Section 2 defines this in a precise
sense.) For example, in team theory, it is common to develop a
strategy that can be implemented by every employee in a given
category and leads to high payoff for the company. (Notice that
this does not lead to identical behavior , because strategies are state-
dependent.) In civic contexts, symmetry commonly arises through
notions of fairness and justice. In treaty negotiations and legislation
that mandates how parties behave, for example, there is often a
constraint that all parties be treated equally. In DecPOMDPs, an
offline solution search may consider only symmetric strategies for
identical agents as a way of reducing the search space. In common-
payoff multiagent reinforcement learning, each agent may collect
percepts and rewards independently, but the reinforcement learning
updates can be pooled to learn a single parameterized policy that
all agents share.
Common-payoff and symmetric games have a number of desir-
able properties that may simplify the search for solutions. For the
Bo
L W
AliL 12
W 01
(a) Bo cannot do laundryBo
L W
AliL 12
W 21
(b) Bo learns to do laundryBo
L W
AliL 32
W 23
(c) Joy in working together
Table 1: Three versions of the laundry/washing up game. Solutions are described in the text.
purposes of this paper, we consider Nash equilibria—strategy pro-
files for all players from which no individual player has an incentive
to deviate—as a reasonable solution concept. For example, Marschak
and Radner [ 20] make the obvious point that a globally optimal
(possibly asymmetric) strategy profile—one that achieves the high-
est common payoff—is necessarily a Nash equilibrium. Moreover,
it can be found in time linear in the size of the payoff matrix.
Another solution concept often used in multiagent RL and dif-
ferential (i.e., continuous-action-space) games is that of a locally
optimal strategy profile—roughly speaking, a strategy profile from
which no player has an incentive to slightly deviate. Obviously, a
locally optimal profile may not be a Nash equilibrium, as a player
may still have an incentive to deviate to some more distant point
in strategy space. Nonetheless, local optima, sometimes called local
Nash equilibria —are important. For example, Ratliff et al .[29] argue
that a local Nash equilibrium may still be stable in a practical sense
if agents are computationally unable to find a better strategy. Simi-
larly, gradient-based game solvers and multiagent RL algorithms
may converge to local optima.
Our first main result, informally stated, is that in a symmetric,
common-payoff game, every local optimum in symmetric strategies is
a (global) Nash equilibrium . Section 3 states the result more precisely
and gives an example illustrating its generality.2Despite many
decades of research on symmetric, common-payoff games, the result
appears to be novel and perhaps useful. There are some echoes of the
result in the literature on single-agent decision making [4, 27, 32],
which can be connected to symmetric solutions of common-payoff
games by treating all players jointly as a single agent, but our result
appears more general than published results. The proof we give of
our result contains elements similar to the proof (of a related but
different result) in Taylor [34].
To gain some intuition for these concepts and claims, let us
consider a situation in which two children, Ali and Bo, have to
do some housework—specifically, laundry ( 𝐿) and washing up ( 𝑊).
Here, the “common payoff,” if any, is to the parents. It is evident
that a symmetric strategy profile—both doing the laundry or both
doing the washing up—is not ideal, because the other task will not
get done.
The first version of the game, whose payoffs 𝑈are shown in
Table 1a, is asymmetric: while Ali is competent at both tasks, Bo
does not know how to do the laundry properly and will ruin the
clothes. Here, as Marschak and Radner pointed out, the strategy
profile(𝐿,𝑊)is both globally optimal and a Nash equilibrium. If
we posit a mixed (randomized) strategy profile in which Ali and Bo
have laundry probabilities 𝑝and𝑞respectively, the gradients 𝜕𝑈/𝜕𝑝
and𝜕𝑈/𝜕𝑞are+1and−1, driving the solution towards (𝐿,𝑊).
2Complete proofs for all of our results are in the appendices.
Figure 1: The strategy profile landscape of the symmetric
laundry game (Figure 1b). Although the symmetric opti-
mum has lower expected utility than the unrestricted op-
tima, total symmetry of the game implies that the symmet-
ric optimum is a Nash equilibrium; this is a special case of
Theorem 3.2.
In the second version of the game (Table 1b), Ali has taught Bo
how to do the laundry, and symmetry is restored. The pure profiles
(𝐿,𝑊)and(𝑊,𝐿)are (asymmetric) globally optimal solutions and
hence Nash equilibria. Figure 1 shows the entire payoff landscape as
a function of 𝑝and𝑞: looking just at symmetric strategy profiles, it
turns out that there is a local optimum at 𝑝=𝑞=0.5, i.e., where Ali
and Bo toss fair coins to decide what to do. Although the expected
payoff of this solution is lower than that of the asymmetric optima,
the local optimum is, nonetheless, a Nash equilibrium. All unilateral
deviations from the symmetric local optimum result in the same
expected payoff because if one child is tossing a coin, the other
child can do nothing to improve the final outcome.
In the third version of the game (Table 1c), the parents derive
greater payoff from watching their children working happily to-
gether on a single task than they do from getting both tasks done. In
this case, there is again a Nash equilibrium at 𝑝=𝑞=0.5, but it is a
local minimum rather than a local maximum in symmetric strategy
space. Thus, not all symmetric Nash equilibria are symmetric local
optima; this is because Nash equilibria depend on unilateral devia-
tions, whereas symmetric local optima depend on joint deviations
that maintain symmetry.
In the second half of the paper, we turn to the issue of robustness
of symmetric solutions. In practice, a variety of factors can lead to
modelling errors and approximate solutions, which motivates us
to consider perturbations in payoffs and strategy profiles. Making
general arguments about Nash equilibria, we show that our first
main result is robust in the sense that it degrades linearly under
𝜖-magnitude perturbations into 𝑘𝜖-Nash equilibria (for some game-
dependent constant 𝑘).
Stability turns out to be a thornier issue. Instability, if not han-
dled carefully, might lead to major coordination failures in practice
[3]. While it is already known that local strict optima in a totally
symmetric team game attain one type of stability, the issue is com-
plex because there are several ways of enforcing (or not enforcing)
strict symmetries in payoffs and strategies [ 22]. Our final results fo-
cus on the stability of agents making possibly-asymmetric updates
from a symmetric solution. We prove for a non-degenerate class
of games that local optima in symmetric strategy space fail to be
local optima in asymmetric strategy space if and only if at least one
player is mixing, and we experimentally quantify how often mixing
occurs for learning algorithms in the GAMUT suite of games [ 24].
2 PRELIMINARIES: GAMES AND
SYMMETRIES
2.1 Normal-form games
Throughout, we consider normal-form games G=(𝑁,𝐴,𝑢)defined
by a finite set 𝑁with|𝑁|=𝑛players, a finite set of action profiles
𝐴=𝐴1×𝐴2×...×𝐴𝑛with𝐴𝑖specifying the actions available to
player𝑖, and the utility function 𝑢=(𝑢1,𝑢2,...,𝑢𝑛)with𝑢𝑖:𝐴→
Rgiving the utility for each player 𝑖[33]. We callGcommon-payoff
if𝑢𝑖(𝑎)=𝑢𝑗(𝑎)for all action profiles 𝑎∈𝐴and all players 𝑖,𝑗. In
common-payoff games we may omit the player subscript 𝑖from
utility functions.
We model each player as employing a (mixed) strategy𝑠𝑖∈
Δ(𝐴𝑖), a probability distribution over actions. We denote the
support of the probability distribution 𝑠𝑖bysupp(𝑠𝑖). Given a
(mixed) strategy profile 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)that specifies a strat-
egy for each player, player 𝑖’s expected utility is 𝐸𝑈𝑖(𝑠)=Í
𝑎∈𝐴𝑢𝑖(𝑎)Î𝑛
𝑗=1𝑠𝑗(𝑎𝑗). If a strategy 𝑠𝑖for player𝑖maximizes ex-
pected utility given the strategies 𝑠−𝑖of all the other players, i.e., if
𝑠𝑖∈argmax𝑠′
𝑖∈Δ(𝐴𝑖)𝐸𝑈𝑖(𝑠′
𝑖,𝑠−𝑖), we call𝑠𝑖abest response to𝑠−𝑖. If
each strategy 𝑠𝑖in a strategy profile 𝑠is a best response to 𝑠−𝑖, we
call𝑠aNash equilibrium . A Nash equilibrium 𝑠isstrict if every𝑠𝑖
is the unique best response to 𝑠−𝑖.
Note that, while we have chosen to use the normal-form game
representation for simplicity, normal-form games are highly ex-
pressive. Normal-form games can represent mixed strategies in all
finite games, including games with sequential actions, stochastic
transitions, and partial observation such as imperfect-information
extensive form games with perfect recall, Markov games, and Dec-
POMDPs. To represent a sequential game in normal form, one
simply lets each normal-form action be a complete strategy (con-
tingency plan) accounting for every potential game decision.
2.2 Symmetry in game structure
Our notion of symmetry in game structure is built upon von Neu-
mann and Morgenstern [ 41]’s and borrows notation from Plan [ 28].
The basic building block is a symmetry of a game :Definition 2.1. Call a permutation of player indices 𝜌:
{1,2,...,𝑛}→{ 1,2,...,𝑛}asymmetry of a game Gif, for all strategy
profiles(𝑠1,𝑠2,...,𝑠𝑛), permuting the strategy profile permutes the
expected payoffs:
𝐸𝑈𝜌(𝑖)((𝑠1,𝑠2,...,𝑠𝑛))=𝐸𝑈𝑖((𝑠𝜌(1),𝑠𝜌(2),...,𝑠𝜌(𝑛))),∀𝑖.
Note that, when we speak of a symmetry of a game, we implicitly
assume𝐴𝑖=𝐴𝑗for all𝑖,𝑗with𝜌(𝑖)=𝑗so that permuting the
strategy profile is well-defined.3
We characterize the symmetric structure of a game by its set of
game symmetries:
Definition 2.2. Denote the set of all symmetries of a game Gby:
Γ(G)={𝜌:{1,2,...,𝑛}→{ 1,2,...,𝑛}a symmetry ofG}.
A spectrum of game symmetries is possible. On one end of the
spectrum, the identity permutation might be the only symmetry
for a given game. On the other end of the spectrum, all possible
permutations might be symmetries for a given game. Following the
terminology of von Neumann and Morgenstern [ 41], we call the
former case totally unsymmetric and the latter case totally symmet-
ric:
Definition 2.3. IfΓ(G)=𝑆𝑛, the full symmetric group, we call
the game Γ(G)totally symmetric . IfΓ(G)contains only the identity
permutation, we call the game totally unsymmetric .
LetP⊆Γ(G)be any subset of the game symmetries. Because
Γ(G)is closed under composition, we can repeatedly apply permu-
tations inPto yield a group of game symmetries ⟨P⟩:
Definition 2.4. LetP⊆Γ(G)be a subset of the game symmetries.
The group generated byP, denoted⟨P⟩, is the set of all permuta-
tions that can result from (possibly repeated) composition of permu-
tations inP:⟨P⟩={𝜌1◦𝜌2◦...◦𝜌𝑚|𝑚∈N,𝜌1,𝜌2,...,𝜌𝑚∈P} .
Group theory tells us that ⟨P⟩defines a closed binary operation
(permutation composition) including an identity and inverse maps,
and⟨P⟩is the closure ofPunder function composition.
With a subset of game symmetries P⊆Γ(G)in hand, we can
use the permutations in Pto carry one player index to another. For
each player 𝑖, we give a name to the set of player indices to which
permutations inPcan carry𝑖: we call it player 𝑖’sorbit.
Definition 2.5. LetP⊆Γ(G)be a subset of the game symmetries
Γ(G). The orbit of player𝑖underPis the set of all other player
indices that⟨P⟩can assign to 𝑖:P(𝑖)={𝜌(𝑖)|𝜌∈⟨P⟩} .
In fact, it is a standard result from group theory that the orbits
of a group action on a set partition the set’s elements, which leads
to the following proposition:
Proposition 2.6. LetP⊆ Γ(G). The orbits ofPpartition the
game’s players.
By Proposition 2.6, each P⊆Γ(G)yields an equivalence relation
among the players. To gain intuition for this equivalence relation,
consider two extreme cases. In a totally unsymmetric game, Γ(G)
contains only the identity permutation, in which case each player
3We make this choice to ease notational burden, but we conjecture that our results
can be generalized to allow for mappings between actions [ 14], which we leave for
future work.
is in its own orbit of Γ(G); the equivalence relation induced by the
orbit partition shows that no players are equivalent. In a totally
symmetric game, by contrast, every permutation is an element of
Γ(G), i.e.,Γ(G)=𝑆𝑛, the full symmetric group; now, all the players
share the same orbit of Γ(G), and the equivalence relation induced
by the orbit partition shows that all the players are equivalent.
We leverage the orbit structure of an arbitrary P ⊆ Γ(G)to
define an equivalence relation among players because it adapts to
however much or little symmetry is present in the game. Between
the extreme cases of no symmetry ( 𝑛orbits) and total symmetry (1
orbit) mentioned above, there could be any intermediate number of
orbits ofP. Furthermore, two players can share an orbit of Peven
if those two players cannot be arbitrarily swapped. In Example 3.3,
all the players can be rotated in a circle, so all the players share an
orbit ofP=Γ(G)even though the game does not admit arbitrary
swapping of players.
2.3 Symmetry in strategy profiles
Having formalized a symmetry of a game in the preceding section,
we follow Nash [ 23] and define symmetry in strategy profiles with
respect to symmetry in game structure:
Definition 2.7. LetP⊆Γ(G)be a subset of the game symmetries
Γ(G). We call a strategy profile 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)P-invariant if
(𝑠1,𝑠2,...,𝑠𝑛)=(𝑠𝜌(1),𝑠𝜌(2),...,𝑠𝜌(𝑛))for all𝜌∈⟨P⟩ .
The equivalence relation among players induced by the orbit
structure ofPis fundamental to our definition of symmetry in
strategy profiles by the following proposition:
Proposition 2.8. A strategy profile 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)isP-invariant
if and only if 𝑠𝑖=𝑠𝑗for each pair of players 𝑖and𝑗withP(𝑖)=P(𝑗).
To state Proposition 2.8 another way, a strategy profile is P-
invariant if all pairs of players 𝑖and𝑗that are equivalent under the
orbits ofPplay the same strategy.
3 LOCAL SYMMETRIC OPTIMA ARE
(GLOBAL) NASH EQUILIBRIA
After the formal definitions of symmetry in the previous section,
we are almost ready to formally state the first of our three main
results. The only remaining definition is that of a local symmetric
optimum:
Definition 3.1. Call𝑠alocally optimalP-invariant strategy profile
of a common-payoff game if: (i) 𝑠isP-invariant, and (ii) for some
𝜖>0, noP-invariant strategy 𝑠′with𝐸𝑈(𝑠′)>𝐸𝑈(𝑠)can be
formed by adding or subtracting at most 𝜖to the probability of
taking any given action 𝑎𝑖∈𝐴𝑖. If, furthermore, condition (ii) holds
for all𝜖>0, we call𝑠aglobally optimalP-invariant strategy profile
or simply an optimalP-invariant strategy profile.
Now we can formally state our first main theorem, that local
symmetric optima are (global) Nash equilibria:
Theorem 3.2. LetGbe a common-payoff normal-form game, and
letP⊆Γ(G)be a subset of the game symmetries Γ(G). Any locally
optimalP-invariant strategy profile is a Nash equilibrium.
Proof. We provide a sketch here and full details in Appendix A.
Suppose, for the sake of contradiction, that an individual player𝑖could beneficially deviate to action 𝑎𝑖(if a beneficial deviation
exists, then there is one to a pure strategy). Then, consider instead
a collective change to a symmetric strategy profile in which all the
players in𝑖’s orbit shift very slightly more probability to 𝑎𝑖. By
making the amount of probability shifted ever smaller, the proba-
bility that this change affects exactly one agent’s realized action
(making it𝑎𝑖when it would not have been before) can be arbitrarily
larger than the probability that it affects multiple agents’ realized
actions. Moreover, if this causes exactly one agent’s realized action
to change, this must be in expectation beneficial, since the origi-
nal unilateral deviation was in expectation beneficial. Hence, the
original strategy profile cannot have been locally optimal. □
3.1 Example illustrating general symmetry
Here, we give an example that shows how Theorem 3.2 is more
general than the case of total symmetry. The example illustrates
the existence of rotational symmetry without total symmetry, and
it illustrates how picking different P ⊆ Γ(G)leads to different
optimalP-invariant strategies and thus different P-invariant Nash
equilibria by Theorem 3.2.
Example 3.3. There are four radio stations positioned in a square.
We number these 1,2,3,4 clockwise, such that, e.g., 1 neighbors 4
and 2. There is also a neighborhood of people at each vertex of the
square. The people can tune in to the radio station at their vertex
of the square and to the radio stations at adjacent vertices of the
square, but they cannot tune in to the station at the opposite vertex.
The game has each radio station choose what to broadcast. For
simplicity, suppose each radio station can broadcast the weather or
music. The common payoff of the game is the sum of the utilities
of the four neighborhoods. For each neighborhood, if the neighbor-
hood cannot tune in to the weather, the payoff for that neighbor-
hood is 0. If the neighborhood can only tune in to the weather, the
payoff is 1, and if the neighborhood can tune in to both weather
and music, the neighborhood’s payoff is 2.
The symmetries of the game Γ(G) include the set of
permutations generated by rotating the radio stations
once clockwise. In standard notation for permutations,
{(1,2,3,4),(2,3,4,1),(3,4,1,2),(4,1,2,3)}⊂Γ(G).
First, consider applying the theorem to P=Γ(G). In this case,
the constraint ofP-invariance requires all the radio stations to play
the same strategy because all stations are in the same orbit. As we
show in Appendix B, the optimal P-invariant strategy then is for
each station to broadcast music with probability√
2−1. Theorem 3.2
tells us that this optimal P-invariant strategy profile is a Nash
equilibrium. Appendix B also shows how to verify this without the
use of Theorem 3.2.
Second, consider applying the theorem to the case where P
consists only of the rotation twice clockwise, i.e., the permutation
which maps each station onto the station on the opposite vertex of
the square. In standard notation for permutations, P={(3,4,1,2)}.
Now, the constraint of P-invariance requires radio stations at op-
posite vertices of the square to play the same strategy. However,
neighboring stations can broadcast different programs. The optimal
P-invariant strategy is for one pair of opposite-vertex radio sta-
tions, e.g., 1 and 3, to broadcast the weather and for the other pair
of radio stations, 2 and 4, to broadcast music. While it turns out to
be immediate that this optimal P-invariant strategy is a Nash equi-
librium because it achieves the globally optimal outcome, we could
have applied Theorem 3.2 to know that this optimal P-invariant
strategy profile is a Nash equilibrium even without knowing what
the optimalP-invariant strategy was .
4 ROBUSTNESS OF THE MAIN RESULT TO
PAYOFF AND STRATEGY PERTURBATIONS
The first type of robustness we consider is robustness to pertur-
bations in the game’s payoff function. Formally, we define an 𝜖-
perturbation of a game as follows:
Definition 4.1. LetGbe a normal-form game with utility function
𝜇. For some𝜖>0, we callG′an𝜖-perturbation ofGifG′has utility
function𝜇′satisfying max𝑖∈𝑁,𝑎∈𝐴|𝑢′
𝑖(𝑎)−𝑢𝑖(𝑎)|≤𝜖.
There are a variety of reasons why 𝜖-perturbations might arise in
practice. Our game model may contain errors such as the game not
being perfectly symmetric; the players’ preferences might drift over
time; or we might have used function approximation to learn the
game’s payoffs. With Proposition 4.2, we note a generic observation
about Nash equilibria showing that our main result, Theorem 3.2, is
robust in the sense of degrading linearly in the payoff perturbation’s
size:
Proposition 4.2. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P⊆Γ(G). Suppose𝐺′is an𝜖-perturbation
ofG. Then𝑠∗is a2𝜖-Nash equilibrium in G′.
The second type of robustness we consider is robustness to sym-
metric solutions that are only approximate. For example, we might
try to find a symmetric local optimum through an approximate
optimization method, or the evolutionary dynamics among play-
ers’ strategies might lead them to approximate local symmetric
optima. Again, a generic result about Nash equilibria shows that
the guarantee of Theorem 3.2 degrades linearly in this case:
Theorem 4.3. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P⊆Γ(G). Suppose𝑠is a strategy profile
with total variation distance 𝑇𝑉(𝑠,𝑠∗) ≤𝛿. Then𝑠is an𝜖-Nash
equilibrium with 𝜖=4𝛿max𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|.
By Theorem 4.3, we have a robustness guarantee in terms of the
total variation distance between an approximate local symmetric
optimum and a true local symmetric optimum. Without much diffi-
culty, we can also convert this into a robustness guarantee in terms
of the Kullback-Leibler divergence:
Corollary 4.4. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P ⊆ Γ(G). Suppose𝑠is a strategy
profile with Kullback-Leibler divergence satisfying 𝐷𝐾𝐿(𝑠||𝑠∗) ≤
𝜈or𝐷𝐾𝐿(𝑠∗||𝑠) ≤𝜈. Then𝑠is an𝜖-Nash equilibrium with 𝜖=
2√
2𝜈max𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|.
While the results of this section show the robustness of Nash
equilibria, we note that Nash equilibria, by definition, consider the
possibility of only a single agent deviating; Nash equilibria cannot
guarantee stability under dynamics that allow for multiple agentsto deviate. In the next section, we investigate when multiple agents
might have an incentive to simultaneously deviate by studying the
optimality of symmetric strategy profiles in possibly-asymmetric
strategy space.
5 WHEN ARE LOCAL OPTIMA IN
SYMMETRIC STRATEGY SPACE ALSO
LOCAL OPTIMA IN
POSSIBLY-ASYMMETRIC STRATEGY
SPACE?
Our first main theoretical result, Theorem 3.2, applies to locally
optimalP-invariant, i.e., symmetric, strategy profiles. This still
leaves open the question of how well locally optimal symmetric
strategy profiles perform when considered in the broader, possibly-
asymmetric strategy space. When are locally optimal P-invariant
strategy profiles also locally optimal in possibly-asymmetric strat-
egy space? This question is important in machine learning (ML)
applications where users of symmetrically optimal ML systems
might be motivated to make modifications to the systems, even for
purposes of a common payoff.
To address this issue more precisely, we formally define a local
optimum in possibly-asymmetric strategy space :
Definition 5.1. A strategy profile 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)of a common-
payoff normal-form game is locally optimal among possibly-
asymmetric strategy profiles , or, equivalently, a local optimum in
possibly-asymmetric strategy space , if for some 𝜖>0, no strategy
profile𝑠′with𝐸𝑈(𝑠′)>𝐸𝑈(𝑠)can be formed by changing 𝑠in
such a way that the probability of taking any given action 𝑎𝑖∈𝐴𝑖
for any player 𝑖changes by at most 𝜖.
Definition 5.1 relates to notions of stability under dynamics, such
as those with perturbations or stochasticity, that allow multiple
players to make asymmetric deviations. In particular, if 𝑠is not a
local maximum in asymmetric strategy space, this means that there
is some set of players 𝐶and strategy 𝑠′
𝐶arbitrarily close to 𝑠, such
that if players 𝐶were to play 𝑠′
𝐶(by mistake or due to stochasticity),
some Player 𝑖∈𝑁−𝐶would develop a strict preference over the
support of𝑠𝑖. To illustrate this, we return to the laundry/washing
up game of the introduction.
Example 5.2. Consider again the game of Table 1b. As Figure 1
illustrates, the symmetric optimum is for both Ali and Bo to random-
ize uniformly between W and L. While this is a Nash equilibrium,
it is not a local optimum in possibly-asymmetric strategy space. If
one player deviates from uniformly randomizing, the other player
develops a strict preference for either 𝑊or𝐿.
To understand when the phenomenon of Example 5.2 happens
in general, we use the following degeneracy condition:
Definition 5.3. Let𝑠be a Nash equilibrium of a game G:
•If𝑠is deterministic, i.e., if every 𝑠𝑖is a Dirac delta function
on some𝑎𝑖, then𝑠isdegenerate if at least two players 𝑖are
indifferent between 𝑎𝑖and some other 𝑎′
𝑖∈𝐴𝑖−{𝑎𝑖}.
•Otherwise, if 𝑠is mixed, then 𝑠isdegenerate if for all players
𝑖and all𝑎−𝑖⊆supp(𝑠−𝑖), the term𝐸𝑈𝑖(𝑎𝑖,𝑎−𝑖)is constant
across𝑎𝑖∈supp(𝑠𝑖).
Intuitively, our definition says that a deterministic Nash equilib-
rium is non-degenerate when it is strict or almost strict (allowing
the exception of at most one player who may be indifferent over
available actions). A mixed Nash equilibrium, on the other hand, is
non-degenerate when mixing matters . When speaking of a game
G, we determine its degeneracy by the degeneracy of its Nash
equilibria:
Definition 5.4. We call a gameGdegenerate if it has at least one
degenerate Nash equilibrium; otherwise, we call Gnon-degenerate .
We note that “degnerate” is already an established term in the
game-theoretical literature where it is often applied only to two-
player games [see, e.g, 42, Definition 3.2]. While similar to the
established notion of degeneracy, our definition of degeneracy is
stronger, which makes our statements about non-degenerate games
more general. If a two-player game Gis non-degenerate in the
usual sense from the literature, it is non-degenerate in the sense
of Definition 5.3. Moreover, if Gis common-payoff, then for each
player𝑖, we can define a two-player game played by 𝑖and another
single player who controls the strategies of 𝑁−{𝑖}. If for all𝑖these
two-player games are non-degenerate in the established sense, then
Gis non-degenerate in the sense of Definition 5.3.
In non-degenerate games, our next theorem shows that a local
symmetric optimum is a local optimum in possibly-asymmetric
strategy space if and only if it is deterministic. Formally:
Theorem 5.5. LetGbe a non-degenerate common-payoff normal-
form game, and let P⊆ Γ(G)be a subset of the game symmetries
Γ(G). A locally optimal P-invariant strategy profile is locally opti-
mal among possibly-asymmetric strategy profiles if and only if it is
deterministic.
To see why the (non-)degeneracy condition is needed in Theo-
rem 5.5, we provide an example of a degenerate game:
Example 5.6. Consider the 3x3 symmetric game with the follow-
ing payoff matrix:
Player 2
a b c
Player 1a1 1 1
b1 -10 1+𝜖
c11+𝜖 -10
Here,(𝑎,𝑎)is the unique global optimum in symmetric strategy
space. By Theorem 3.2, it is therefore also a Nash equilibrium. How-
ever, it is a degenerate Nash equilibrium and not locally optimal
in asymmetric strategic space. The payoff can be improved by, e.g.,
Player 1 playing 𝑏with small probability (and 𝑎otherwise) and
Player 2 playing 𝑐with small probability (and 𝑎otherwise).
The following game illustrates how a global symmetric optimum,
even if it is a non-degenerate, deterministic equilibrium, might still
not be globally optimal in possibly-asymmetric strategy space.
Example 5.7. Consider|𝑁|=3and𝐴={0,1}, let𝑘be the
number of players who choose action 1, and let the payoffs be: 0if
𝑘=0,−1if𝑘=1,1if𝑘=2, and−1if𝑘=3.
Then the global symmetric optimum is for everyone to play 0.
The global asymmetric optimum, on the other hand, is to coordi-
nate to achieve 𝑘=2. Hence, the global symmetric optimum isstrictly worse than the global asymmetric optimum. Of course, by
Theorem 5.5,(0,0,0)is still a local optimum of asymmetric strategy
space.
6 LEARNING SYMMETRIC STRATEGIES IN
GAMUT
Theorem 5.5 shows that, in non-degenerate games, a locally optimal
symmetric strategy profile is stable in the sense of Section 5 if and
only if it is pure. For those concerned about stability, this raises the
question: how often are optimal strategies pure, and how often are
they mixed?
To answer this question, we present an empirical analysis of
learning symmetric strategy profiles in the GAMUT suite of game
generators [ 24]. We are interested both in how centralized optimiza-
tion algorithms (such as gradient methods) search for symmetric
strategies and in how decentralized populations of agents evolve
symmetric strategies. To study the former, we run Sequential Least
SQuares Programming (SLSQP) [ 17,40], a local search method
for constrained optimization. To study the latter, we simulate the
replicator dynamics [ 9], an update rule from evolutionary game
theory with connections to reinforcement learning [ 2,36,37]. (See
Appendix E.3 for details.)
6.1 Experimental setup
We ran experiments in all three classes of symmetric GAMUT
games: RandomGame, CoordinationGame, and CollaborationGame.
Intuitively, a RandomGame draws all payoffs uniformly at random,
whereas in a CoordinationGame and a CollaborationGame, the high-
est payoffs are always for outcomes where all players choose the
same action. (See Appendix E.1 details.) Because CoordinationGame
and CollaborationGame have such similar game structures, our ex-
perimental results in the two games are nearly identical. To avoid
redundancy, we only include experimental results for Coordina-
tionGame in this paper.
For each game class, we sweep the parameters of the game from
2 to 5 players and 2 to 5 actions, i.e., with (|𝑁|,|𝐴𝑖|)∈{ 2,3,4,5}×
{2,3,4,5}. We sample 100 games at each parameter setting and
then attempt to calculate the global symmetric optimum using
(i) 10 runs of SLSQP and (ii) 10 runs of the replicator dynamic
(each with a different initialization drawn uniformly at random
over the simplex), resulting in 10 + 10 = 20 solution attempts per
game. Because we do not have ground truth for the globally optimal
solution of the game, we instead use the best of our 20 solution
attempts, which we call the “best solution.”
To apply our previously developed theory to GAMUT games,
we observe that RandomGames, CoordinationGames, and Collab-
orationGames are (almost surely) non-degenerate in the sense of
Definition 5.4:
Proposition 6.1. Drawing a degenerate game is a measure-zero
event in RandomGames, CoordinationGames, and CollaborationGames.
6.2 What fraction of symmetric optima are
local optima in possibly-asymmetric
strategy space?
Here, we try to get a sense for how often symmetric optima are
stable in the sense that they are also local optima in possibly-
asymmetric strategy space (see Section 5). In Appendix Table 3b,
we show in what fraction of games the best solution of our 20 opti-
mization attempts is mixed; by Theorem 5.5, this is the fraction of
games whose symmetric optima are not local optima in possibly-
asymmetric strategy space. In CoordinationGames, the symmetric
optimum is always (by construction) for all players to choose the
same action, leading to stability. By contrast, we see that 36% to
60% of RandomGames are unstable . We conclude that if real-world
games do not have the special structure of CoordinationGames,
then instability may be common.
6.3 How often do SLSQP and the replicator
dynamic find an optimal solution?
As sequential least squares programming and the replicator dy-
namic are not guaranteed to converge to a global optimum, we test
empirically how often each run converges to the best solution of
our 20 optimization runs. In Appendix Table 4 / Table 6, we show
what fraction of the time any single SLSQP / replicator dynamics
run finds the best solution, and in Appendix Table 5 / Table 7, we
show what fraction of the time at least 1 of 10 SLSQP / replicator
dynamics runs finds the best solution. First, we note that the tables
for SLSQP and the replicator dynamics are quite similar, differing by
no more than a few percentage points in all cases. So the replicator
dynamics, which are used as a model for how populations evolve
strategies, can also be used as an effective optimization algorithm.
Second, we see that individual runs of each algorithm are up to
93% likely to find the best solution in small RandomGames, but
they are less likely (as little as 24% likely) to find the best solution
in larger RandomGames and in CoordinationGames. The best of
10 runs, however, finds the best solution ≥87%of the time, indi-
cating that random algorithm restarts benefit symmetric strategy
optimization.
7 COMPUTATIONAL COMPLEXITY OF
COMPUTING GAME SYMMETRIES AND
SYMMETRIC STRATEGIES
7.1 Finding symmetries
In some cases, domain knowledge can provide the symmetries of
a game. For example, in the laundry game of Table 1b, symmetry
arises from a simple observation: it matters only what chores get
done, not which children do the chores. In other cases, however,
players may face a potentially symmetric common-payoff game
and first have to determine what the symmetries of the game are,
e.g., by computing a generating set of the group of symmetries.
Call this problem the game automorphism (GA) problem. Can it be
solved efficiently?
The complexity of the GA problem depends on how the game
is represented. The simplest representation is to give the full table
of payoffs. However, the size is then exponential in the number of
players. A simple alternative is to only explicitly represent non-zeroentries in the payoff table. This way, some games of many players
can be represented succinctly. Calling the latter a sparse represen-
tation and the former a non-sparse representation, we obtain the
following:
Theorem 7.1. On a non-sparse game representation, the GA prob-
lem can be solved in polynomial time. On a sparse representation, the
GA problem is polynomial-time equivalent to the graph isomorphism
problem.
For a general introduction to the graph isomorphism problem,
see Grohe and Schweitzer [11]. Notably, the problem is in NP but
neither known to be solvable in polynomial time nor known to be
NP-hard.
7.2 Finding an optimal symmetric strategy
profile
Once it is known what the symmetries Pof a given game are, what
is the complexity of finding an optimal P-invariant strategy profile?
In Appendix G.2, we show that the problem of optimizing symmetric
strategies is equivalent to the problem of optimizing polynomials
on Cartesian products of unit simplices. However, depending on
how the polynomials and games are represented, the reductions
may increase the problem instance exponentially. Nevertheless, we
can import results from the literature on optimizing polynomials
to obtain results such as the following:
Theorem 7.2. Deciding for a given game Gwith symmetriesP
and a given number 𝐾whether there is aP-invariant profile with
expected utility at least 𝐾is NP-hard, even for 2-player symmetric
games.
8 CONCLUSION
When ML is deployed in the world, it is natural to instantiate multi-
ple agents from the same template. This naturally restricts strategy
profiles to symmetric ones, and it puts the focus on finding optimal
symmetric strategy profiles. This, in turn, raises questions about the
properties of such profiles. Would individual agents (or the users
they serve) want to deviate from these profiles? Are they robust
to small changes in the game or in the executed strategies? Could
there be better asymmetric strategy profiles nearby?
Our results yield a mix of good and bad news. Theorems 3.2 and
4.3 are good news, showing that even local optima in symmetric
strategy space are (global) Nash equilibria in a robust sense. So,
with respect to unilateral deviations among team members, sym-
metric optima are relatively stable strategies. On the other hand,
Theorem 5.5 is perhaps bad news, because it shows that a broad
class of symmetric local optima are unstable when considering joint
deviations in asymmetric strategy space (Section 5). Furthermore,
our empirical results with learning algorithms in GAMUT suggest
that these unstable solutions may not be uncommon in practice
(Section 6.2).
REFERENCES
[1]V. Arvind, Bireswar Das, Johannes Köbler, and Seinosuke Toda. 2010. Colored
Hypergraph Isomorphism is Fixed Parameter Tractable. In Proceedings of the
IARCS International Conference on Foundations of Software Technology and Theo-
retical Computer Science . Dagstuhl Publishing, 327–337. https://doi.org/10.4230/
LIPIcs.FSTTCS.2010.327
[2]Tilman Börgers and Rajiv Sarin. 1997. Learning through reinforcement and
replicator dynamics. Journal of economic theory 77, 1 (1997), 1–14.
[3]Nick Bostrom, Thomas Douglas, and Anders Sandberg. 2016. The Unilateralist’s
Curse and the Case for a Principle of Conformity. Social epistemology 30, 4 (2016),
350–371.
[4]Rachael Briggs. 2010. Putting a value on Beauty. In Oxford Studies in Epistemology .
Vol. 3. Oxford University Press, 3–24.
[5]Constantinos Daskalakis and Christos H. Papadimitriou. 2007. Computing Equi-
libria in Anonymous Games. In FOCS .
[6]Etienne de Klerk. 2008. The complexity of optimizing over a simplex, hypercube
or sphere: a short survey. Central European Journal of Operations Research 16
(2008), 111–125. https://link.springer.com/article/10.1007/s10100-007-0052-9
[7]Jakob Foerster, Ioannis Alexandros Assael, Nando De Freitas, and Shimon White-
son. 2016. Learning to communicate with deep multi-agent reinforcement learn-
ing. In Advances in neural information processing systems . 2137–2145.
[8]Jakob N Foerster, Gregory Farquhar, Triantafyllos Afouras, Nantas Nardelli, and
Shimon Whiteson. 2018. Counterfactual multi-agent policy gradients. In Thirty-
second AAAI conference on artificial intelligence .
[9]Drew Fudenberg and David K Levine. 1998. The theory of learning in games . Vol. 2.
MIT press.
[10] Joaquim Gabarró, Alina García, and Maria Serna. 2011. The complexity of game
isomorphism. Theoretical Computer Science 412, 48 (11 2011), 6675–6695. https:
//doi.org/10.1016/j.tcs.2011.07.022
[11] Martin Grohe and Pascal Schweitzer. 2020. The Graph Isomorphism Problem.
Commun. ACM 63, 11 (11 2020), 128–134. https://doi.org/10.1145/3372123
[12] Jayesh K Gupta, Maxim Egorov, and Mykel Kochenderfer. 2017. Cooperative multi-
agent control using deep reinforcement learning. In International Conference on
Autonomous Agents and Multiagent Systems . Springer, 66–83.
[13] Dylan Hadfield-Menell, Anca D. Dragan, Pieter Abbeel, and Stuart J. Russell. 2017.
Cooperative inverse reinforcement learning. In Advances in Neural Information
Processing 29 .
[14] Nicholas Ham. 2013. Notions of Symmetry for Finite Strategic-Form Games.
arXiv:1311.4766.
[15] Ramesh Johari. 2007. Fictitious play: Examples and convergence. URL: http:
//web.stanford.edu/~rjohari/teaching/notes/336\_lecture7\_2007.pdf.
[16] Matthias Köppe. 2010. On the Complexity of Nonlinear Mixed-Integer Optimization .
https://arxiv.org/pdf/1006.4895.pdf Also published in https://link.springer.com/
chapter/10.1007/978-1-4614-1927-3\_19.
[17] Dieter Kraft. 1988. A Software Package for Sequential Quadratic Programming .
Technical Report DFVLR-FB 88-28. DLR German Aerospace Center – Institute
for Flight Mechanics, Koln, Germany.
[18] Eugene M. Luks. 1999. Hypergraph Isomorphism and Structural Equivalence
of Boolean Functions. In STOC ’99: Proceedings of the thirty-first annual ACM
symposium on Theory of Computing . Atlanta, GA, USA, 652–658.
[19] Jakob Marschak. 1955. Elements for a theory of teams. Management Science 1, 2
(1955), 127–137.
[20] Jacob Marschak and Roy Radner. 1972. Economic Theory of Teams . Yale University
Press.
[21] Rudolf Mathon. 1979. A note on the graph isomorphism counting problem. Inform.
Process. Lett. 8, 3 (3 1979), 131–136. https://doi.org/10.1016/0020-0190(79)90004-8
[22] Igal Milchtaich. 2016. Static stability in symmetric and population games . Technical
Report. https://www.biu.ac.il/soc/ec/wp/2008-04/2008-04.pdf
[23] John Nash. 1951. Non-cooperative games. Annals of Mathematics (1951), 286–295.
[24] E Nudelman, J Wortman, Y Shoham, and K Leyton-Brown. 2004. Run the GAMUT:
a comprehensive approach to evaluating game-theoretic algorithms. In Proceed-
ings of the Third International Joint Conference on Autonomous Agents and Multi-
agent Systems, 2004. AAMAS 2004. IEEE, 880–887.
[25] Frans A Oliehoek, Christopher Amato, et al .2016. A Concise Introduction to
Decentralized POMDPs . Springer.
[26] Asu Ozdaglar. 2010. Learning in Games. URL: https://ocw.mit.edu/courses/
electrical-engineering-and-computer-science/6-254-game-theory-with-
engineering-applications-spring-2010/lecture-notes/MIT6\_254S10\_lec11.pdf.
[27] Michele Piccione and Ariel Rubinstein. 1997. On the interpretation of decision
problems with imperfect recall. Games and Economic Behavior 20, 1 (1997), 3–24.
[28] Asaf Plan. 2017. Symmetric n-player games. Working paper, available at asaf-
plan.com.
[29] Lillan J. Ratliff, Samuel A. Burden, and S. Shankar Sastry. 2016. On the Charac-
terization of Local Nash Equilibria in Continuous Games. IEEE Trans. Automat.
Control 61, 8 (Aug. 2016), 2301–2307. https://doi.org/10.1109/TAC.2016.2583518
[30] Philip J Reny. 1999. On the existence of pure and mixed strategy Nash equilibria
in discontinuous games. Econometrica 67, 5 (1999), 1029–1056.
[31] Stuart Russell. 2019. Human Compatible: Artificial Intelligence and the Problem of
Control . Viking Press.
[32] Wolfgang Schwarz. 2015. Lost memories and useless coins: revisiting the absent-
minded driver. Synthese 192 (2015), 3011–3036.
[33] Yoav Shoham and Kevin Leyton-Brown. 2008. Multiagent Systems: Algorithmic,
Game-Theoretic, and Logical Foundations . Cambridge University Press. http:
//www.masfoundations.org/[34] Jessica Taylor. 2016. In memoryless Cartesian environments, every UDT
policy is a CDT+SIA policy. https://www.alignmentforum.org/posts/
5bd75cc58225bf06703751b2/in-memoryless-cartesian-environments-every-
udt-policy-is-a-cdt-sia-policy
[35] Alexandre Tsybakov. 2009. Introduction to Nonparametric Estimation . Springer.
[36] Karl Tuyls, Dries Heytens, Ann Nowe, and Bernard Manderick. 2003. Extended
replicator dynamics as a key to reinforcement learning in multi-agent systems.
InEuropean Conference on Machine Learning . Springer, 421–431.
[37] Karl Tuyls, Katja Verbeeck, and Tom Lenaerts. 2003. A selection-mutation model
for q-learning in multi-agent systems. In Proceedings of the second international
joint conference on Autonomous agents and multiagent systems . 693–700.
[38] Takashi Ui. 2009. Bayesian potentials and information structures: Team decision
problems revisited. International Journal of Economic Theory 5, 3 (2009), 271–291.
[39] Steen Vester. 2012. Symmetric Nash Equilibria . Ph.D. Dissertation.
[40] Pauli Virtanen, Ralf Gommers, Travis E. Oliphant, Matt Haberland, Tyler
Reddy, David Cournapeau, Evgeni Burovski, Pearu Peterson, Warren Weckesser,
Jonathan Bright, Stéfan J. van der Walt, Matthew Brett, Joshua Wilson, K. Jar-
rod Millman, Nikolay Mayorov, Andrew R. J. Nelson, Eric Jones, Robert Kern,
Eric Larson, C J Carey, İlhan Polat, Yu Feng, Eric W. Moore, Jake VanderPlas,
Denis Laxalde, Josef Perktold, Robert Cimrman, Ian Henriksen, E. A. Quintero,
Charles R. Harris, Anne M. Archibald, Antônio H. Ribeiro, Fabian Pedregosa,
Paul van Mulbregt, and SciPy 1.0 Contributors. 2020. SciPy 1.0: Fundamental Al-
gorithms for Scientific Computing in Python. Nature Methods 17 (2020), 261–272.
https://doi.org/10.1038/s41592-019-0686-2
[41] John von Neumann and Oskar Morgenstern. 1944. Theory of Games and Economic
Behavior . Princeton University Press.
[42] Bernhard von Stengel. 2007. Equilibrium Computation for Two-Player Games in
Strategic and Extensive Form. In Algorithmic Game Theory . Cambridge University
Press.
A PROOFS OF SECTION 3 RESULTS
Theorem 3.2. LetGbe a common-payoff normal-form game, and
letP⊆Γ(G)be a subset of the game symmetries Γ(G). Any locally
optimalP-invariant strategy profile is a Nash equilibrium.
Proof. We proceed by contradiction. Suppose 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)
is locally optimal among P-invariant strategy profiles that is not a
Nash equilibrium. We will construct an 𝑠′arbitrarily close to 𝑠with
𝐸𝑈(𝑠′)>𝐸𝑈(𝑠).
Without loss of generality, suppose 𝑠1is not a best response to
𝑠−1but that the pure strategy of always playing 𝑎1is a best response
to𝑠−1. For an arbitrary probability 𝑝>0, consider the modified
strategy𝑠′
1that plays action 𝑎1with probability 𝑝and follows 𝑠1
with probability 1−𝑝. Now, construct 𝑠′=(𝑠′
1,𝑠′
2,...,𝑠′𝑛)as follows:
𝑠′
𝑖=(
𝑠′
𝑖=𝑠′
1if𝑖∈P( 1)
𝑠′
𝑖=𝑠𝑖otherwise.
In words,𝑠′modifies𝑠by having the members of player 1’s orbit
mix in a probability 𝑝of playing𝑎1. We claim for all sufficiently
small𝑝that𝐸𝑈(𝑠′)>𝐸𝑈(𝑠).
To establish this claim, we break up the expected utility of 𝑠′
according to cases of how many players in 1’s orbit play the action
𝑎1because of mixing in 𝑎1with probability 𝑝. In particular, we
observe
𝐸𝑈(𝑠′)=𝐵(𝑚=0,𝑝)𝐸𝑈(𝑠)
+𝐵(𝑚=1,𝑝)𝐸𝑈((𝑠′
1,𝑠2,...,𝑠𝑛))
+𝐵(𝑚>1,𝑝)𝐸𝑈(...),
where𝐵(𝑚,𝑝)is the probability of 𝑚successes for a binomial ran-
dom variable on 𝑚independent events that each have success
probability𝑝and where𝐸𝑈(...)is arbitrary. Note that the crucial
step in writing this expression is grouping the terms with the coef-
ficient𝐵(𝑚=1,𝑝). We can do this because for any player 𝑗∈P( 1),
there exists a symmetry 𝜌∈Γ(G)with𝜌(𝑗)=1.
Now, to achieve 𝐸𝑈(𝑠′)>𝐸𝑈(𝑠), we require
𝐸𝑈(𝑠)<𝐵(𝑚=1,𝑝)
𝐵(𝑚>0,𝑝)𝐸𝑈((𝑠′
1,𝑠2,...,𝑠𝑛))
+𝐵(𝑚>1,𝑝)
𝐵(𝑚>0,𝑝)𝐸𝑈(...).
We know𝐸𝑈((𝑠′
1,𝑠2,...,𝑠𝑛))>𝐸𝑈(𝑠), but we must deal with the
case when𝐸𝑈(...)is arbitrarily negative. Because lim𝑝→0𝐵(𝑚>
1,𝑝)/𝐵(𝑚=1,𝑝)=0, by making 𝑝sufficiently small, 𝐵(𝑚=
1,𝑝)/𝐵(𝑚>0,𝑝)can be made greater than 𝐵(𝑚>1,𝑝)/𝐵(𝑚>
0,𝑝)by an arbitrarily large ratio. The result follows. □
B OPTIMAL SYMMETRIC POLICY FOR THE
RADIO STATION GAME OF EXAMPLE 3.3
We here calculate the optimal Γ(G)-invariant strategy profile for
Example 3.3. Let 𝑝be the probability of broadcasting the weather
forecasts. By symmetry of the game and linearity of expectation,
the expected utility given 𝑝is simply four times the expected utility
of any individual neighborhood. The value of an individual neigh-
borhood is 0with probability(1−𝑝)3, is1with probability 𝑝3and
is2with the remaining probability. Hence, the expected utility of asingle neighborhood is
𝑝3+(1−(1−𝑝)3−𝑝3)·2=2−2(1−𝑝)3−𝑝3.
The maximum of this term (and thus the maximum of the overall
utility of all neighborhoods) can be found by any computer algebra
system to be 𝑝=2−√
2, which gives an expected utility of 4(√
2−
1)≈1.66.
To double-check, we can also calculate the symmetric Nash equi-
librium of this game. It’s easy to see that that Nash equilibrium
must be mixed and therefore must make each player (radio station)
indifferent about what to broadcast. So let 𝑝again be the probability
with which everyone else broadcasts the weather. The expected
utility of broadcasting the weather relative to broadcasting nothing
for any of the three relevant neighborhoods is 2(1−𝑝)2. (Broadcast-
ing the weather lifts the utility of a neighborhood from 0 to 2 if they
do not already get the weather. Otherwise, it is useless to air the
weather.) The expected utility of broadcasting music again relative
to broadcasting nothing is simply 𝑝2. We can find the symmetric
Nash equilibrium by setting
2(1−𝑝)2=𝑝2,
which gives us the same solution for 𝑝as before.
C PROOFS OF SECTION 4 RESULTS
Proposition 4.2. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P⊆Γ(G). Suppose𝐺′is an𝜖-perturbation
ofG. Then𝑠∗is a2𝜖-Nash equilibrium in G′.
Proof. By Theorem 3.2, 𝑠∗is a Nash equilibrium in G. After
perturbingGby𝜖to formG′, payoffs have increased / decreased
at most±𝜖, so the difference between any two actions’ expected
payoffs has changed by at most 2𝜖. □
Theorem 4.3. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P⊆Γ(G). Suppose𝑠is a strategy profile
with total variation distance 𝑇𝑉(𝑠,𝑠∗) ≤𝛿. Then𝑠is an𝜖-Nash
equilibrium with 𝜖=4𝛿max𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|.
Proof. Consider the perspective of an arbitrary player 𝑖. The
difference in expected utility of playing any action 𝑎𝑖between the
opponent strategy profiles 𝑠−𝑖and𝑠∗
−𝑖is given by:
𝐸𝑈𝑖(𝑎𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)
=Õ
𝑎−𝑖∈𝐴−𝑖𝑠−𝑖(𝑎−𝑖)𝑢𝑖(𝑎𝑖,𝑎−𝑖)
−Õ
𝑎−𝑖∈𝐴−𝑖𝑠∗
−𝑖(𝑎−𝑖)𝑢𝑖(𝑎𝑖,𝑎−𝑖)
≤Õ
𝑎−𝑖∈𝐴−𝑖|𝑢𝑖(𝑎𝑖,𝑎−𝑖)|𝑠−𝑖(𝑎−𝑖)−𝑠∗
−𝑖(𝑎−𝑖)
≤2𝑇𝑉(𝑠,𝑠∗)max
𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|
≤2𝛿max
𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|.
In particular, let 𝑎𝑖be an action in the support of 𝑠∗
𝑖, and let𝑎′
𝑖
be any other action. Then, using the above, we have:
𝐸𝑈𝑖(𝑎′
𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠−𝑖)
=𝐸𝑈𝑖(𝑎′
𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎′
𝑖,𝑠∗
−𝑖)+𝐸𝑈𝑖(𝑎′
𝑖,𝑠∗
−𝑖)
−𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)+𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠−𝑖)
≤𝐸𝑈𝑖(𝑎′
𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎′
𝑖,𝑠∗
−𝑖)
+𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠−𝑖)
≤𝐸𝑈𝑖(𝑎′
𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎′
𝑖,𝑠∗
−𝑖)
+𝐸𝑈𝑖(𝑎𝑖,𝑠−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)
≤4𝛿max
𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|,
where𝐸𝑈𝑖(𝑎′
𝑖,𝑠∗
−𝑖)−𝐸𝑈𝑖(𝑎𝑖,𝑠∗
−𝑖)≤0because𝑠∗
𝑖is a Nash equilib-
rium by Theorem 3.2. □
Corollary 4.4. LetGbe a common-payoff normal-form game,
and let𝑠∗be a locally-optimal P-invariant strategy profile for some
subset of game symmetries P ⊆ Γ(G). Suppose𝑠is a strategy
profile with Kullback-Leibler divergence satisfying 𝐷𝐾𝐿(𝑠||𝑠∗) ≤
𝜈or𝐷𝐾𝐿(𝑠∗||𝑠) ≤𝜈. Then𝑠is an𝜖-Nash equilibrium with 𝜖=
2√
2𝜈max𝑖∈𝑁,𝑎∈𝐴|𝑢𝑖(𝑎)|.
Proof. By Pinsker’s inequality [35], we have
𝑇𝑉(𝑠,𝑠∗)≤r
1
2𝐷𝐾𝐿(𝑠||𝑠∗).
As𝑇𝑉(𝑠,𝑠∗)=𝑇𝑉(𝑠∗,𝑠)and with a similar application of Pinsker’s
inequality, we have by assumption that 𝑇𝑉(𝑠,𝑠∗)≤p
𝜈/2. Applying
Theorem 4.3 with 𝛿=p
𝜈/2yields the result. □
D PROOFS OF SECTION 5 RESULTS
Theorem 5.5. LetGbe a non-degenerate common-payoff normal-
form game, and let P⊆ Γ(G)be a subset of the game symmetries
Γ(G). A locally optimal P-invariant strategy profile is locally opti-
mal among possibly-asymmetric strategy profiles if and only if it is
deterministic.
Proof. Let𝑠be a locally optimal P-invariant strategy profile. By
Theorem 3.2, 𝑠is a Nash equilibrium. Because Gis non-degenerate,
so is𝑠. We prove the claim by proving that (1) if 𝑠is deterministic, it
is locally optimal in asymmetric strategy space; and (2) if 𝑠is mixed
then it is not locally optimal in asymmetric strategy space.
(1) The deterministic case: Let 𝑠be deterministic. Now consider
a potentially asymmetric strategy profile 𝑠′. We must show as 𝑠′
becomes sufficiently close to 𝑠that𝐸𝑈(𝑠′)≤𝐸𝑈(𝑠).
Let𝜖1,𝜖2,...,𝜖𝑛and ˆ𝑠1,...,ˆ𝑠𝑛be such that for 𝑖∈𝑁,𝑠′
𝑖can be
interpreted as following 𝑠𝑖with probability 1−𝜖𝑖and following ˆ𝑠𝑖
with probability 𝜖𝑖, where𝑠𝑖∉supp(ˆ𝑠𝑖). Then (similar to the proofof Theorem 3.2), we can write
𝐸𝑈(𝑠′)
= Ö
𝑖∈𝑁(1−𝜖𝑖)!
𝐸𝑈(𝑠)
+Õ
𝑗∈𝑁𝜖𝑗©
«Ö
𝑖∈𝑁−{𝑗}1−𝜖𝑖ª®
¬·𝐸𝑈(ˆ𝑠𝑗,𝑠−𝑗)
+Õ
𝑗,𝑙∈𝑁:𝑗≠𝑙𝜖𝑗𝜖𝑖©
«Ö
𝑖∈𝑁−{𝑗,𝑙}1−𝜖𝑖ª®
¬·𝐸𝑈(ˆ𝑠𝑗,ˆ𝑠𝑙,𝑠−𝑗−𝑙)
+...
The second line is the expected value if everyone plays 𝑠, the third
line is the sum over the possibilities of one player 𝑗deviating to
ˆ𝑠𝑗, and so forth. We now make two observations. First, because
𝑠is a Nash equilibrium, the expected utilities 𝐸𝑈(ˆ𝑠𝑗,𝑠−𝑗)in the
third line are all at most as big as 𝐸𝑈(𝑠). Now consider any later
term corresponding to the deviation of some set 𝐶, containing at
least two players 𝑖,𝑗. Note that it may be 𝐸𝑈(ˆ𝑠𝐶,𝑠−𝐶)>𝐸𝑈(𝑠).
However, this term is multiplied by 𝜖𝑖𝜖𝑗. Thus, as the 𝜖go to 0, the
significance of this term in the average vanishes in comparison
to that of both the terms corresponding to the deviation of just 𝑖
and just𝑗, which are multiplied only by 𝜖𝑖and𝜖𝑗, respectively. By
non-degeneracy, it is 𝐸𝑈(ˆ𝑠𝑖,𝑠−𝑖)<𝐸𝑈(𝑠)or𝐸𝑈(ˆ𝑠𝑗,𝑠−𝑗)<𝐸𝑈(𝑠).
Thus, if the 𝜖𝑖are small enough, the overall sum is less than 𝐸𝑈(𝑠).
(2) The mixed case: Let 𝑠be mixed. We proceed by constructing a
strategy profile 𝑠′that is arbitrarily close to 𝑠with𝐸𝑈(𝑠′)>𝐸𝑈(𝑠).
Let𝑚be the largest integer where for all subsets of players 𝐶⊆𝑁
with|𝐶|≤𝑚, the expected payoff is constant across all joint devia-
tions to𝑎𝑖∈supp(𝑠𝑖)for all𝑖∈𝐶, i.e., where𝐸𝑈(𝑎𝐶,𝑠−𝐶)=𝐸𝑈(𝑠)
for all𝑎𝐶∈supp(𝑠𝐶). As𝑠is a non-degenerate Nash equilibrium,
1≤𝑚<𝑛.
By definition of 𝑚, there exists a subset of players 𝐶⊂𝑁with
|𝐶|=𝑚and choice of actions 𝑎𝐶∈supp(𝑠𝐶)where𝐸𝑈(𝑎𝑗,𝑎𝐶,𝑠−𝑗−𝐶)
is not constant across the available actions 𝑎𝑗∈𝐴𝑗for some player
𝑗∈𝑁−𝐶. Denote player 𝑗’s best response to the joint deviation 𝑎𝐶
as𝑎∗
𝑗∈argmax𝑎𝑗𝐸𝑈(𝑎𝑗,𝑎𝐶,𝑠−𝑗−𝐶), and note𝐸𝑈(𝑎𝑗,𝑎𝐶,𝑠−𝑗−𝐶)>
𝐸𝑈(𝑎𝐶,𝑠−𝐶)=𝐸𝑈(𝑠).
To construct 𝑠′, modify𝑠by letting player 𝑗mix according to
𝑠𝑗with probability(1−𝜖)and play action 𝑎𝑗with probability 𝜖.
Similarly, let each player 𝑖∈𝐶mix according to 𝑠𝑖with probability
(1−𝜖)and play their action 𝑎𝑖specified by 𝑎𝐶with probability 𝜖.
Because we allow 𝜖>0to be arbitrarily small, all we have left to
show is𝐸𝑈(𝑠′)>𝐸𝑈(𝑠).
Observe as before that we can break 𝐸𝑈(𝑠′)up into cases based
on the number of players who deviate according to the modified
Player 2
𝛼 𝛽
Player 1𝛼𝑢𝛼𝛼𝑢𝛼𝛽
𝛽𝑢𝛼𝛽𝑢𝛽𝛽
Table 2: A payoff matrix with |𝑁|=2and𝐴1=𝐴2={𝛼,𝛽}
to illustrate GAMUT games. In a RandomGame, 𝑢𝛼𝛼,𝑢𝛼𝛽,
and𝑢𝛽𝛽are i.i.d. draws from Unif(−100,100). In a Coordi-
nationGame, 𝑢𝛼𝛼and𝑢𝛽𝛽are i.i.d. draws from Unif(0,100)
while𝑢𝛼𝛽is a draw from Unif(−100,0). In a Collabora-
tionGame, 𝑢𝛼𝛼=𝑢𝛽𝛽=100, and𝑢𝛼𝛽is a draw from
Unif(−100,99).
probability𝜖:
𝐸𝑈(𝑠′)
=©
«Ö
𝑘∈𝐶∪{𝑗}(1−𝜖)ª®
¬𝐸𝑈(𝑠)
+Õ
𝑙∈𝐶∪{𝑗}𝜖©
«Ö
𝑘∈𝐶∪{𝑗}:𝑘≠𝑙1−𝜖ª®
¬𝐸𝑈(𝑎𝑙,𝑠−𝑙)
+...
+©
«Ö
𝑘∈𝐶∪{𝑗}𝜖ª®
¬𝐸𝑈(𝑎𝑗,𝑎𝐶,𝑠−𝑗−𝐶).
By construction, every value in the expected value calculation
𝐸𝑈(𝑠′)is equal to𝐸𝑈(𝑠)except for the last value 𝐸𝑈(𝑎𝑗,𝑎𝐶,𝑠−𝑗−𝐶),
which is greater than 𝐸𝑈(𝑠). We conclude 𝐸𝑈(𝑠′)>𝐸𝑈(𝑠).□
E GAMUT DETAILS AND ADDITIONAL
EXPERIMENTS
E.1 GAMUT games
In Section 6.1, we analyzed all three classes of symmetric GAMUT
games: RandomGame, CoordinationGame, and CollaborationGame.
Below, we give a formal definiton of these game classes:
Definition E.1. ARandomGame with|𝑁|players and|𝐴|actions
assumes𝐴𝑖=𝐴𝑗for all𝑖,𝑗and draws a payoff from Unif(−100,100)
for each unordered action profile 𝑎∈𝐴.
Definition E.2. ACoordinationGame with|𝑁|players and|𝐴|
actions assumes 𝐴𝑖=𝐴𝑗for all𝑖,𝑗. For each unordered action
profile𝑎∈𝐴with𝑎𝑖=𝑎𝑗for all𝑖,𝑗, it draws a payoff from
Unif(0,100); for all other unordered action profiles, it draws a
payoff from Unif(−100,0).
Definition E.3. ACollaborationGame with|𝑁|players and|𝐴|
actions assumes 𝐴𝑖=𝐴𝑗for all𝑖,𝑗. For each unordered action
profile𝑎∈𝐴with𝑎𝑖=𝑎𝑗for all𝑖,𝑗, the payoff is 100; for all other
unordered action profiles, it draws a payoff from Unif(−100,99).
Note that these games define payoffs for each unordered action
profile because the games are totally symmetric (Definition 2.3).
Table 2 gives illustrative examples.
Figure 2: The magnitude of the replicator dynamics update
step averaged over 10,000 RandomGames4with 2 players and
2 actions. Although this plot indicates that the replicator dy-
namics converge by 100 iterations, we ran 10,000 iterations
for good measure in all of our experiments.
E.2 Proof of Proposition 6.1
Proof. By Definition 5.4, in order for a game to be degenerate,
there must exist a player 𝑖, a set of actions for the other players
𝑎−𝑖, and a pair of actions 𝑎𝑖≠𝑎′
𝑖with𝐸𝑈(𝑎𝑖,𝑎−𝑖)=𝐸𝑈(𝑎′
𝑖,𝑎−𝑖).
In RandomGames, CoordinationGames, and CollaborationGames,
𝐸𝑈(𝑎𝑖,𝑎−𝑖)=𝜇(𝑎𝑖,𝑎−𝑖)and𝐸𝑈(𝑎′
𝑖,𝑎−𝑖)=𝜇(𝑎′
𝑖,𝑎−𝑖)are continu-
ous random variables that are independent of each other. (Or, in
the case of a CollaborationGame, 𝜇(𝑎𝑖,𝑎−𝑖)may be a fixed value
outside of the support of 𝜇(𝑎′
𝑖,𝑎−𝑖).) So𝐸𝑈(𝑎𝑖,𝑎−𝑖)=𝐸𝑈(𝑎′
𝑖,𝑎−𝑖)
is a measure-zero event. □
E.3 Replicator dynamics
Consider a game where all players share the same action set, i.e.,
with𝐴𝑖=𝐴𝑗for all𝑖,𝑗, and consider a totally symmetric strategy
profile𝑠=(𝑠1,𝑠1,...,𝑠 1). In the replicator dynamic, each action
can be viewed as a species, and 𝑠1defines the distribution of each
individual species (action) in the overall population (of actions).
At each iteration of the replicator dynamic, the prevalence of an
individual species (action) grows in proportion to its relative fitness
in the overall population (of actions). In particular, the replicator
dynamic evolves 𝑠1(𝑎)over time𝑡for each𝑎∈𝐴1as follows:
𝑑
𝑑𝑡𝑠1(𝑎)=𝑠1(𝑎)[𝐸𝑈(𝑎,𝑠−1)−𝐸𝑈(𝑠)].
To simulate the replicator dynamic with Euler’s method, we need to
choose a stepsize and a total number of iterations. Experimentally,
we found the fastest convergence with a stepsize of 1, and we found
that 100 iterations sufficed for convergence; see Figure 2. For good
measure, we ran 10,000 iterations of the replicator dynamic in all
of our experiments.
4In this simulation only we rescaled the RandomGames so that each payoff is a draw
from Unif(0,1).
We are interested in the replicator dynamic for two reasons.
First, it is a model for how agents in the real world may collectively
arrive at a symmetric solution to a game (e.g., through evolutionary
pressure). Second, it is a learning algorithm that performs local
search in the space of symmetric strategies. In our experiments of
Appendix E.5, we find that using the replicator dynamic as an opti-
mization algorithm is competitive with Sequential Least SQuares
Programming (SLSQP), a local search method from the constrained
optimization literature [17, 40].
E.4 What fraction of symmetric optima are
local optima in possibly-asymmetric
strategy space?
As discussed in Section 6.2, we would like to get a sense for how
often symmetric optima are stable in the sense that they are also
local optima in possibly-asymmetric strategy space (see Section 5).
Table 3 shows in what fraction of games the best solution we found
isunstable .
E.5 How often do SLSQP and the replicator
dynamic find an optimal solution?
As discussed in Section 6.3, Table 4 and Table 5 show how often
SLSQP finds an optimal solution, while Table 6 and Table 7 show
how often the replicator dynamic finds an optimal solution.
E.6 How costly is payoff perturbation under
the simultaneous best response dynamic?
When a game is not stable in the sense of Section 5, we would like to
understand how costly the worst-case 𝜖-perturbation of the game
can be. (See Definition 4.1 for the definition of an 𝜖-perturbation of
a game.) In particular, we study the case when individuals simul-
taneously update their strategies in possibly-asymmetric ways by
defining the following simultaneous best response dynamic :
Definition E.4. The simultaneous best response dynamic at 𝑠up-
dates from strategy profile 𝑠=(𝑠1,𝑠2,...,𝑠𝑛)to strategy profile
𝑠′=(𝑠′
1,𝑠′
2,...,𝑠′𝑛)with every𝑠′
𝑖a best response to 𝑠−𝑖.
For each of the RandomGames in Section 6.2 whose symmetric
optimum𝑠is not a local optimum in possibly-asymmetric strat-
egy space, we compute the worst-case 𝜖payoff perturbation for
infinitesimal 𝜖. Then, we update each player’s strategy according
to the simultaneous best response dynamic at 𝑠. This necessarily
leads to a decrease in the original common payoff because the play-
ers take simultaneous updates on an objective that, after payoff
perturbation, is no longer common. Table 8 reports the average
percentage decrease in expected utility, which ranges from 55% to
89%. Our results indicate that simultaneous best responses after
payoff perturbation in RandomGames can be quite costly.
F THE COMPUTATIONAL COMPLEXITY OF
FINDING THE SYMMETRIES OF A GAME
In this section, we analyze the computational complexity of finding
the symmetries of a common-payoff game. In general, symmetries
as defined in Definition 2.1 can be found in exponential time in the
number of players. Therefore, if we represent the game explicitlyas a full payoff matrix, then the symmetries can be found in poly-
nomial time in the size of the input. However, if we can represent
the game more efficiently by giving only non-zero entries of the
payoff matrix, the problem becomes graph isomorphism-complete,
i.e., polynomial-time equivalent to the graph isomorphism prob-
lem, which is neither known to be solvable in polynomial time
nor known to be NP-hard [see 11, for an overview]. We also show
(in Section F.3) that if we consider a more general notion of game
symmetry that permutes actions in addition to players, the com-
putational problem becomes graph isomorphism-complete on an
explicit payoff matrix representation.
F.1 The hypergraph automorphism problem
We here introduce the hypergraph automorphism problem and
some existing results about it. In the next section, we will prove
our results by relating the game automorphism problem (i.e., the
problem of finding the symmetries of a game) to the hypergraph
automorphism problem.
Ahypergraph is a pair(𝑉,𝐸), where𝑉is a (finite) set of vertices
and𝐸⊆2𝑉is a set of hyperedges .
Asymmetry or automorphism of a hypergraph is a bijection
𝜌:𝑉→𝑉s.t. for each set of vertices 𝑒⊆𝑉, it is𝑒∈𝐸if and
only if𝜌(𝑒)∈𝐸, where𝜌(𝑒)B{𝜌(𝑣)|𝑣∈𝑒}. In other words:
For𝜌to be a symmetry it must be the case that any set of vertices
𝑣1,...,𝑣𝑘are connected by a hyperedge if and only if 𝜌(𝑣1),...,𝜌(𝑣𝑘)
are connected by a hyperedge.
Definition F.1. The hypergraph automorphism (HA) problem asks
for a given hypergraph (𝑉,𝐸)to provide a set of symmetries of
(𝑉,𝐸)that generate the group of all symmetries of (𝑉,𝐸).
There are two natural ways to represent the edges of a hyper-
graph. The first is to provide what one would call an adjacency
matrix in the case of a regular graph. That is, we give a table of bits
that specifies for each 𝑒∈2𝑉whether𝑒∈𝐸. That is, for each set of
vertices, we specify whether there is a hyperedge connecting that
set of vertices. The downside of this representation style is that it
always costs 𝑂(2|𝑉|)bits. An alternative is to explicitly list 𝐸, such
that graphs with few edges can be represented in much less space
than𝑂(2|𝑉|). We call the former notation non-sparse and the latter
sparse .
Lemma F.2. The HA problem on a sparse hypergraph representa-
tion is graph isomorphism-complete.
Proof. Mathon [ 21] shows that the problem of giving the gen-
erators of the automorphism group of a given graph isomorphism
complete. So it is left to show that the automorphism problem on
sparse hypergraphs is polynomially equivalent to analogous prob-
lem on graphs. Since graphs are hypergraphs, we only need to
reduce HA to the graph automorphism problem. This is easy and
the main idea has been noted before, e.g., see the introduction of
Arvind et al. [1]. □
Theorem F.3 (Luks, 1999, Theorem 4.2). The HA problem is solv-
able in𝑂(𝑐|𝑉|)for some constant 𝑐. In particular, it follows immedi-
ately that HA on non-sparse representations is solvable in polynomial
time.
A 2 3 4 5
N
2 0.36 0.44 0.44 0.50
3 0.38 0.49 0.59 0.60
4 0.42 0.45 0.46 0.46
5 0.45 0.48 0.49 0.47
(a) RandomGameA 2 3 4 5
N
2 0 0 0 0
3 0 0 0 0
4 0 0 0 0
5 0 0 0 0
(b) Coordina-
tionGame
Table 3: The fraction of games whose symmetric optima are mixed. By Theorem 5.5, these symmetric equilibria are the ones
unstable in the sense of Section 5. Numbers in the table were empirically determined from 100 randomly sampled games per
GAMUT class.
A 2 3 4 5
N
2 0.92 0.81 0.70 0.64
3 0.80 0.69 0.57 0.48
4 0.75 0.57 0.40 0.35
5 0.70 0.45 0.36 0.31
(a) RandomGameA 2 3 4 5
N
2 0.59 0.50 0.40 0.33
3 0.53 0.38 0.28 0.29
4 0.53 0.37 0.29 0.26
5 0.53 0.36 0.33 0.25
(b) CoordinationGame
Table 4: The fraction of single SLSQP runs that achieve the best solution found in our 20 total optimization attempts. Numbers
in the table were empirically determined from 100 randomly sampled games per GAMUT class.
A 2 3 4 5
N
2 1.00 0.99 0.99 0.98
3 1.00 0.99 1.00 0.96
4 1.00 0.96 0.94 0.88
5 0.98 0.90 0.88 0.91
(a) RandomGameA 2 3 4 5
N
2 0.99 1.00 0.98 0.97
3 1.00 0.99 0.93 0.95
4 1.00 0.97 0.97 0.93
5 0.99 1.00 0.95 0.92
(b) CoordinationGame
Table 5: The fraction of games in which at least 1 of 10 SLSQP runs achieves the best solution found in our 20 total optimization
attempts. Numbers in the table were empirically determined from 100 randomly sampled games per GAMUT class.
A 2 3 4 5
N
2 0.93 0.81 0.68 0.65
3 0.81 0.70 0.58 0.46
4 0.76 0.58 0.36 0.34
5 0.69 0.43 0.36 0.30
(a) RandomGameA 2 3 4 5
N
2 0.58 0.45 0.40 0.33
3 0.57 0.35 0.29 0.27
4 0.53 0.37 0.28 0.25
5 0.51 0.33 0.33 0.24
(b) CoordinationGame
Table 6: The fraction of single replicator dynamics runs that achieve the best solution found in our 20 total optimization
attempts. Numbers in the table were empirically determined from 100 randomly sampled games per GAMUT class.
This is harder to show. Note that a brute force method that tests
all|𝑉|!bijections is super-exponential in |𝑉|and super-polynomial
in (the problem size) 2|𝑉|.
A 2 3 4 5
N
2 1.00 1.00 1.00 1.00
3 0.99 1.00 0.95 0.96
4 1.00 0.98 0.91 0.91
5 0.98 0.97 0.92 0.87
(a) RandomGameA 2 3 4 5
N
2 1.00 1.00 0.99 0.94
3 1.00 0.97 0.93 0.96
4 0.99 1.00 0.93 0.92
5 1.00 0.98 0.96 0.90
(b) CoordinationGame
Table 7: The fraction games in which at least 1 of 10 replicator dynamics runs achieves the best solution found in our 20 total
optimization attempts. Numbers in the table were empirically determined from 100 randomly sampled games per GAMUT
class.
A 2 3 4 5
N
2 58.9% 55.9% 61.8% 64.6%
3 73.7% 70.9% 73.4% 73.7%
4 74.1% 77.4% 78.4% 82.5%
5 77.4% 84.9% 89.9% 87.5%
(a) RandomGame
Table 8: The average decrease in expected utility that worst-case infinitesimal asymmetric payoff perturbations cause to un-
stable symmetric optima. To get these numbers, we first perturb payoffs in the 100 RandomGames from Section 6.2 whose
symmetric optima 𝑠are not local optima in possibly-asymmetric strategy space. Then, in each perturbed game, we compute a
simultaneous best-response update to 𝑠and record its decrease in expected utility.
F.2 Polynomial-time equivalence
Recall from the main text that – for the purpose of our paper – a
symmetry of an 𝑛-player (common-payoff) game (𝐴,𝑢)is a per-
mutation𝜌:{1,...,𝑛}→{ 1,...,𝑛}s.t. for all pure strategy profiles
a∈𝐴, it is𝑢(𝑎1,...,𝑎𝑛)=𝑢(𝑎𝜌(1),...,𝑎𝜌(𝑛)). In particular, we do
not consider permutations of the actions.
Definition F.4. The (common-payoff) game automorphism (GA)
problem asks us to compute for a common-payoff given game a
generating set of the symmetries of the game.
As with hypergraphs, we distinguish two representations for a
game. A sparse representation lists only non-zero payoffs. A non-
sparse representation gives the payoff for each 𝐴. As before, the
downside of the full payoff table representation is that its size is
exponential in the number of players.
Theorem F.5. The sparse/non-sparse representation HA problem
is polynomial-time equivalent to the sparse/non-sparse representation
GA problem.
By polynomial-time, we here mean in time bound by a poly-
nomial in the number of players and the size in bits of the given
instance.
Proof. HA→GA: We first reduce the HA problem to the GA
problem, which is the easier direction. We will use the same con-
struction for both the sparse-to-sparse and non-sparse-to-non-
sparse case.
Take a given hypergraph (𝑉,𝐸). WLOG assume 𝑉={1,...,𝑛}.
Then we construct an 𝑛-player game, in which each player has twoactions,𝑎0,𝑎1. For any𝑀⊆{1,...,𝑛}, leta𝑀be the payoff profile
in which the set of players playing 𝑎1is exactly𝑀. Then let the
payoff𝑢(a𝑀)be1if𝑀∈𝐸and0otherwise.
We now show that this reduction is valid by showing that the
game and the graph have the same symmetries. Let 𝜌be a bijection.
Then:
𝜌is a symmetry of(𝑉,𝐸) (1)
iff∀𝑒∈2𝑉:𝑒∈𝐸⇐⇒𝜌(𝑒)∈𝐸 (2)
iff∀𝑀∈2𝑉:𝑢(a𝑀)=1⇐⇒𝑢(a𝜌(𝑀))=1 (3)
iff∀𝑀∈2𝑉:𝑢(a𝑀)=𝑢(a𝜌(𝑀)) (4)
iff∀a∈𝐴:𝑢(𝑎1,...,𝑎𝑛)=𝑢(𝑎𝜌(1),...,𝑎𝜌(𝑛)) (5)
iff𝜌is a symmetry of(𝐴,𝑢) (6)
It is easy to see that this construction can be performed in poly-
nomial (indeed linear) time for both sparse and non-sparse repre-
sentations.
GA→HA: We now reduce in the opposite direction. This is more
complicated and we therefore provide only a sketch.
Consider an 𝑛-player game(𝐴,𝑢). We construct the hypergraph
as follows. First, for each player 𝑖, we generate a vertex. We also
generate log2(|𝐴𝑖|)vertices that we use to encode 𝐴𝑖, player𝑖’s
actions and connect them with the vertex representing 𝑖. For players
𝑖,𝑗that have the same action label sets, this encoding must be done
consistently for 𝑖,𝑗.
We also need to add some kind of structure to ensure that sym-
metries of the hypergraph can only map the 𝑘th action-encoding
vertex of player 𝑖on the𝑘th action-encoding vertex of a player 𝑗
that has the same action label set as 𝑖.
Next, we represent the payoff function 𝑢. To do so, we introduce
⌈log2(|𝑢(𝐴)−{ 0}|)⌉payoff encoding vertices. Note that |𝑢(𝐴)−
{0}|is the number of distinct non-zero payoffs of the game. To
encode|𝑢(𝐴)−{ 0}|we therefore need ⌈log2(|𝑢(𝐴)−{ 0}|)⌉bits.
We connect these bits in such a way that any symmetry must
map each of them onto itself. We fix some binary encoding of
𝑢(𝐴)−{ 0}. For instance, let’s say the non-zero payoffs of the game
are{−3,−1,7,8,10,11,13,100}. Then we need three bits, and might
encode them as−3↦→000,−1↦→001,7↦→010,8↦→011, and so
forth.
For each a∈𝐴with𝑢(a)≠0, we then add an edge that contains
for each Player 𝑖the action encoding vertices corresponding to
𝑎𝑖; and those bits from the payoff encoding vertices that together
represent the payoff. (So, for example, if the payoff is encoded by
011, then the hyperedge contains the two lower payoff encoding
vertices. Similarly for the action encoding vertices.)
We omit a proof of the correctness of this reduction.
It is left to show that the reduction is polynomial-time for both
representation styles. For the sparse representation styles, it is triv-
ial because up to some small number of extra vertices and edges,
there is a one-to-one correspondence between edges of the hyper-
graph and action profiles with non-zero payoffs.
On to the non-sparse representation. Clearly each entry of the
adjacency matrix can be filled in polynomial (perhaps even constant
or logarithmic) time. It is left to show that the adjacency matrix is
not too large. In particular, we need to show that the size of the
adjacency matrix is polynomial in the size of the payoff matrix.
To assess the size of the adjacency matrix we need to count the
number of vertices in the above construction. First, the number of
player vertices is
𝑛≤log2(|𝐴1|)+...+log2(|𝐴𝑛|)
=log2(|𝐴1|·...·|𝐴𝑛|)
=log2(|𝐴|).
(The inequality assumes each player has at least two actions.) Sec-
ond, the number of action-encoding vertices is
⌈log2(|𝐴1|)⌉+...+⌈log2(|𝐴𝑛|)⌉
≤2 log2(|𝐴1|)+...+2 log2(|𝐴𝑛|)
=2 log2(|𝐴|).
Finally, the number of payoff encoding vertices is about
⌈log2(|𝑢(𝐴)|)⌉≤ 2 log2(|𝑢(𝐴)|)≤ 2 log2(|𝐴|).
The overall number of vertices in the above construction is therefore
at most 5log2(|𝐴|). Thus, the size (in terms of number of bits) of
the adjacency matrix is bound by
25(log2|𝐴|)=|𝐴|5.
Since|𝐴|is a lower bound on the size of the payoff matrix (in
bits), this is polynomial in the size of the game’s payoff matrix, as
required. □
One might wonder: In the non-sparse representation case, why
does the reduction to HA not also work if we use a more traditional
sense of game symmetries? If it were to work that would show thatGI is polynomial-time solvable! But this does not work (with the
proof strategy used above). In the current reduction, actions do not
get their own vertices. Thus, (even if we dropped the constraint
structures that prevent actions from being remapped), a hypergraph
automorphism cannot remap, e.g., an action encoded as 11011 to
an action encoded as 01010. To express full action relabelings in
the hypergraph, it seems that we need to introduce an action per
vertex. However, the size of the adjacency matrix then blows up
more than polynomially.
Combining Lemma F.2 and Theorems F.3 and F.5, we get a char-
acterization of the complexity of the graph automorphism problem.
Corollary F.6. GA is solvable in polynomial time on a non-sparse
representation and is GI-complete on a sparse representation.
F.3 An alternative notion of game symmetry
As mentioned in the main text, we only consider symmetries that
relabel the players and the above is on the computational problem
resulting from that notion of symmetry. As noted in footnote 3, this
was done in part to keep notation simple and an alternative, slightly
more complicated notion allows for the actions to be permuted. A
natural question then is what the complexity is of finding this new
type of symmetry in a given common-payoff game. In this case,
the answer is that finding symmetries is GI-complete regardless of
how the game is represented and follows almost immediately from
existing ideas from Mathon [21] and Gabarró et al. [10].
A player-action (PA) symmetry of a game is pair of a bijec-
tion𝜌:{1,...,𝑛}→{ 1,...,𝑛}on players and a family of bijections
𝜏𝑖:𝐴𝜌(𝑖)→𝐴𝑖s.t. for all pure strategy profiles (𝑎1,...,𝑎𝑛),
𝑢(𝑎1,...,𝑎𝑛)=𝑢(𝜏1(𝑎𝜌(1)),...,𝜏𝑛(𝑎𝜌(𝑛))).
So the idea in this new definition is that 𝜏𝑖translates the action
names from those of player 𝜌(𝑖)to player𝑖.
Define the PAGA problem analogously to the GA problem above,
as finding a generating set of the PA symmetries of a given common-
payoff game. This time, the complexity is independent of whether
the game is represented sparsely or not.
Theorem F.7. The PAGA problem is GI-complete.
Proof. PAGA→GIFor their proof of GI-completeness of the
game isomorphism problem, Gabarró et al . [10] sketch how a general-
sum game can be represented as a graph in a way that maintains
the isomorphisms. In particular, we can therefore represent a sin-
gle common-payoff game as a graph in a way that maintains PA
symmetries. The have thus given a sketch of a polynomial-time
reduction from PAGA to the problem of finding the automorphisms
of a graph. This latter problem can in turn be reduced in polynomial
time to the graph isomorphism problem as was shown by Mathon
[21].
GI→PAGA Second, we show GI-hardness. As shown by Mathon
[21], it is enough to reduce the graph automorphism problem to
the PAGA problem. For this, we can slightly modify the construc-
tion of Gabarró et al . [10, Lemma 5 ]. Specifically, they reduce the
graph isomorphism problem to the 4-player general-sum game
isomorphism problem, where actions represent vertices and the
graph isomorphisms can be recovered from those PA game isomor-
phisms where the player isomorphism is 𝜌=id. Obviously, the
same construction can be used to reduce the graph automorphism
problem to the general-sum game automorphism problem. The only
issue is therefore that their construction uses general-sum games,
but we can simply encode their payoff vectors as single numbers.
In particular, because their payoffs are binary, we might translate
(0,0,0,0)↦→ 0,(0,0,0,1)↦→ 1,(0,0,1,0)↦→ 2, and so forth. It’s easy
to see that the symmetries with 𝜌=idremain the same under this
transformation. □
G THE COMPUTATIONAL COMPLEXITY OF
FINDING OPTIMAL SYMMETRIC
STRATEGIES
G.1 Polynomials
A (multivariate) polynomial in 𝑘variables is a function
(𝑥1,...,𝑥𝑘)↦→Õ
𝑒1,...,𝑒𝑘∈{1,...,𝑚}𝑐(𝑒1,...,𝑒𝑘)𝑥𝑒1
1...𝑥𝑒𝑘
1
for some𝑚, where the 𝑐(𝑒1,...,𝑒𝑘)are some set of real coefficients.
The terms𝑐(𝑒1,...,𝑒𝑘)𝑥𝑒1
1...𝑥𝑒𝑘
1for which𝑐(𝑒1,...,𝑒𝑘)≠0are called
the monomials of the polynomial. The maxdegree of a monomial
𝑐(𝑒1,...,𝑒𝑘)𝑥𝑒1
1...𝑥𝑒𝑘
1ismax𝑖=1,...,𝑘𝑒𝑖. The maxdegree of a polynomial
is the maximum of the maxdegrees of its monomials. Similarly, the
total degree of a monomial 𝑐(𝑒1,...,𝑒𝑘)𝑥𝑒1
1...𝑥𝑒𝑘
1isÍ𝑘
𝑖=1𝑒𝑖. The total de-
gree of a polynomial is the maximum of the total degrees of its mono-
mials. The degree of a variable 𝑥𝑖in a monomial 𝑐(𝑒1,...,𝑒𝑘)𝑥𝑒1
1...𝑥𝑒𝑘
1
is𝑒𝑖. The maxdegree of 𝑥𝑖in a polynomial is the maximum of the
maxdegrees of 𝑥𝑖in all of the monomials.
We can partition the parameters of a polynomial into vectors
and write the polynomial as 𝑓(x1,...,x𝑘), where x1,...,x𝑘are real
vectors. We define the degree of x𝑖in a monomial as the sum of
the degrees of the entries of x𝑖in the monomial. We define the
maxdegree of x𝑖in the polynomial as the maximum of the degrees
ofx𝑖in the polynomial’s monomials.
In the following we will interpret the set Δ(𝐴𝑖)of probability
distributions over 𝐴𝑖as the set of|𝐴𝑖|-dimensional vectors of non-
negative reals whose entries sum to one. We will index these vectors
by𝐴𝑖(rather than numbers 1,...,|𝐴𝑖|). The sets Δ(𝐴𝑖)are also called
unit simplices.
G.2 Optimizing symmetric strategies as
maximizing polynomials
It is immediately obvious that in a symmetric game, the expected
utility as a function of the probabilities that each of the orbits assign
to each of the strategies is a polynomial over a Cartesian product
of unit simplices. Formally:
Proposition G.1. LetGbe an𝑛-player game andP ⊆ Γ(G)
be a subset of the game symmetries of G. Let the orbits ofPbe
𝑀1,...,𝑀𝑘. Further, let the set of actions of orbit 𝑖be𝐴𝑖. Then the
expected utility function over P-invariant strategy profiles of Gis a
polynomial over Δ(𝐴1)×...×Δ(𝐴𝑘)with a max degree of (at most)
max𝑖|𝑀𝑖|and a total degree of (at most) 𝑛. This polynomial can be
created in polynomial-time in the size of a sparse or non-sparse (as
per Appendix F.2) representation of the game.
It follows that we can use algorithms for optimizing polynomials
to find optimal symmetric strategies and that positive results onoptimizing polynomials transfer to finding optimal mixed strategies.
Unfortunately, these results are generally somewhat cumbersome
to state. This is because the optimum can in general not be repre-
sented exactly algebraically, even using 𝑛-th roots, as implied by the
Abel–Ruffini theorem. Positive results must therefore be given in
terms of approximations of the optimal solution. One striking result
from the literature is that, roughly speaking, for a fixed number of
variables, the optimal solution can be approximated in polynomial
time [ 16, Section 6.1]. Translated to our setting, this means that the
optimal symmetric strategy can be approximated in polynomial
time if we keep constant the number of orbits and the number of
actions available to each orbit, but potentially increase the number
of players in each orbit. For more discussion of the complexity of
optimizing polynomials on unit simplices, see de Klerk [6].
G.3 Expressing polynomials as symmetric
games
We now show that, conversely, for any polynomial over a Cartesian
product of simplices there exists a symmetric game whose expected
utility term is exactly that polynomial. However, depending on how
we represent polynomials and how we represent games, the size of
the game may blow up exponentially.
We first show that each polynomial over Δ(𝐴1)×...×Δ(𝐴𝑘)can
be rewritten in such a way that each input x𝑖appears in the same
degree in all monomials.
Lemma G.2. Let𝑓(x1,...,x𝑘)be a polynomial on real vectors of
dimensions𝐴1,...,𝐴𝑘. Then there exists a polynomial 𝑔on the same
inputs s.t. for all(x1,...,x𝑘)∈Δ(𝐴1)×...×Δ(𝐴𝑘)
𝑔(x1,...,x𝑘)=𝑓(x1,...,x𝑘),
and the degree of every x𝑖in all monomials of 𝑔is the maxdegree of
x𝑖in𝑓.
Proof. Consider any monomial ˜𝑓of𝑓in which x𝑖does not have
its max degree. Then for all (x1,...,x𝑘)∈Δ(𝐴1)×...×Δ(𝐴𝑘),
˜𝑓(x1,...,x𝑘)
=©
«Õ
𝑎𝑖∈𝐴𝑖𝑥𝑖,𝑎𝑖ª®
¬˜𝑓(x1,...,x𝑘)
=Õ
𝑎𝑖∈𝐴𝑖𝑥𝑖,𝑎𝑖˜𝑓(x1,...,x𝑘).
Notice that this is the sum of |𝐴𝑖|monomials in which x𝑖occurs
in 1 plus the degree in which it occurs in ˜𝑓. We can iterate this
transformation until we arrive at the desired ˜𝑓. □
Note, however, that if web take a given polynomial represented
as a sum of monomials – e.g., 𝑓(𝑥1,𝑥2)=𝑥4
1−3𝑥2– and rewriting
it as outlined in the Lemma and its proof, the size may blow up
exponentially. E.g., 𝑓(𝑥1,𝑥2)=𝑥4
1−3𝑥2=𝑥4
1−3(𝑥1+𝑥2)3𝑥2
and(𝑥1+𝑥2)3expands into a sum of 23=8terms. However, in
some table-of-coefficient representations of polynomials the size
of the instance does not change at all and the transformation can
be performed in polynomial time in the input. For example, this is
the case if𝑘=1and we represent a polynomial as a table of the
coefficients of all terms 𝑥𝑒1
1...𝑥𝑒𝑘
𝑘where𝑒1+...+𝑒𝑘are at most the
polynomial’s maxdegree.
Once we have a polynomial of the structure described in Lemma G.2,
we can transform it into a game:
Proposition G.3. Let𝑓(x1,...,x𝑘)be a polynomial in which each
x𝑖appears in the same degree in all monomials. Then we can construct
a gameGwith symmetriesPthat create𝑘orbits, where the number
of players in orbit 𝑖=1,...,𝑘 is the degree of x𝑖in𝑓and the number
of actions for the players in orbit 𝑖is the number of entries of x𝑖.
Proof. Consider games Γwith orbits𝑀1,...,𝑀𝑘of the specified
sizes and sets of actions 𝐴1,...,𝐴𝑘also of the specified sizes where
specifically the players in each 𝑀𝑖are totally symmetric. Then such
a game if fully specified as follows. For each family of numbers
𝑛1,1,...,𝑛 1,|𝐴1|,....,𝑛𝑘,1,...,𝑛𝑘,|𝐴𝑘|with𝑛𝑖,1+...+𝑛𝑖,|𝐴𝑖|=|𝑀𝑖|we
need to specify the utility 𝑣obtained if for all 𝑖,𝑙,𝑛𝑖,𝑗players in
orbit𝑖play action number 𝑗from𝐴𝑖.
In the expected utility function of G, each such entry creates a
summand
𝑣· Ö
𝑖|𝑀𝑖|
𝑛𝑖,1,...,𝑛𝑖,|𝐴𝑖|!Ö
𝑖,𝑙𝑝𝑛𝑖,𝑙
𝑖,𝑙,
where𝑝𝑖,𝑙is the probability with which players in orbit 𝑖player
action𝑙and|𝑀𝑖|
𝑛𝑖,1,...,𝑛𝑖,|𝐴𝑖|is a multinomial. By setting 𝑣appropri-
ately, we can thus obtain any monomial with exponents (𝑛𝑖,𝑙)𝑖,𝑙. By
setting the values 𝑣all different sets of (𝑛𝑖,𝑙)𝑖,𝑙appropriately, we
obtain any polynomial in which each (𝑛𝑖,𝑙)𝑙appears with the same
degree|𝑀𝑖|in all monomials. □
Note that if the polynomial is represented as a table of coeffi-
cients, then this reduction takes linear time in the size of the input.
Similarly, if the polynomial is given as a list of only the monomials
with non-zero coefficients – all of which satisfy the degree require-
ment – the reduction can also be done in polynomial time. This in
particular gives us the following negative result, translated from
the literature on optimizing polynomials:
Corollary G.4. Deciding for a given game Gwith symmetries
Pand a given number 𝐾whether there is aP-invariant profile with
expected utility at least 𝐾is NP-hard, even for 2-player symmetric
games.
Proof. Follows from Proposition G.3 and the NP-hardness of
optimizing quadratic polynomials over the unit simplex [ 6, Section
3.2]. □

Scott Emmons
Caspar Oesterheld
Andrew Critch
Vince Conitzer
Stuart Russell