Notions of Anonymity, Fairness and Symmetry for Finite Strategic-Form Games
aa r X i v : . [ m a t h . C O ] A ug Notions of Symmetry for FiniteStrategic-Form Games
Nicholas HamAugust 8, 2018
Abstract
In this paper we survey various notions of symmetry for finite strategic-form games; show thatgame bijections and game isomorphisms form groupoids; introduce matchings as a convenient char-acterisation of strategy triviality; and outline how to construct and partially order parameterisedsymmetric games with numerous examples that range all combinations of surveyed symmetry no-tions.
Keywords
Game Theory, Symmetric Games, Game Automorphisms.
The notion of a game being fair may be made more precise with the concept of symmetry. Broadlyspeaking we will consider a game fair when the players are indifferent between which position they play,however there are several distinct notions of symmetry that are possible which lead to variations instructure and fairness. For example, the players may or may not care about the arrangement of theiropponents.This paper aims to survey numerous notions of symmetry for finite strategic-form games whilst fillingvarious holes. Symmetry in the context of games was first explored by von Neumann and Morgenstern(1944), outlining what we will later refer to as our label-dependent framework in which player permuta-tions act on strategy profiles, consequently requiring all players have the same strategy labels.Soon after Nash (1951) famously showed that symmetric games have at least one mixed strategy Nashequilibrium that is invariant under player permutations, while more recently Cheng et al. (2004) showedthat fully symmetric 2-strategy games have at least one pure strategy Nash equilibrium.Under the theme of anonymity rather than fairness, Brandt et al. (2009) examined label-dependentnotions where players are indifferent between who plays which strategy.Nash (1951), Peleg et al. (1999), Sudh¨olter et al. (2000), and Stein (2011) have examined notions ofsymmetry which may not be captured inside our label-dependent framework. In order to discuss andanalyse such notions we will need to make a detour to examine morphisms between games, the complexityof which has been investigated by Gabarr´o et al. (2007). Inside what will later be referred to as our label-independent framework game automorphisms act on strategy profiles, which also allows players to havedistinct strategy labels.We begin in Section 2 by reviewing numerous mathematical concepts that will play an important rolethroughout our analysis. In Section 3 we survey various label-dependent notions of anonymity andfairness.In Section 4 we review game morphisms while showing that game bijections and game isomorphisms formgroupoids, which appears to be missing from relevant literature, and introduce matchings as a convenientcharacterisation of strategy triviality.
Nicholas HamSchool of Mathematics & Physics, University of Tasmania,Email: [email protected]
In this paper we will only concern ourselves with finite games, consequently all sets are implicitly finite.Let N = { , ..., n } where n ≥ { A i : i ∈ N } be a collection of non-empty sets, with the Cartesianproduct of A , ..., A n denoted as × i ∈ N A i . To simplify notation, for each i ∈ N we denote × j ∈ N −{ i } A j as A − i , for each s ∈ × i ∈ N A i and i ∈ N we denote the element of A i in position i as s i , and for each s i ∈ A i and s − i = ( s , ..., s i − , s i +1 , ..., s n ) ∈ A − i we denote ( s , ..., s i − , s i , s i +1 , ..., s n ) ∈ × i ∈ N A i as ( s i , s − i ).A relation on A , ..., A n is a subset R of their Cartesian product × i ∈ N A i . Let i ∈ N , we say that R is i -total when for each s i ∈ A i there exists s − i ∈ A − i such that ( s i , s − i ) ∈ R , and i -unique when( s i , s − i ) , ( s i , s ′− i ) ∈ R implies s − i = s ′− i .A strategic-form game , or just game when contextually unambiguous, consists of a set N = { , ..., n } of n ≥ i ∈ N , a non-empty set A i of strategies and a utility function u i : A → R , where A denotes the set of strategy profiles × i ∈ N A i . We denote such a game as the triple( N, A, u ), where u = ( u i ) i ∈ N . If there exists m ∈ Z + such that | A i | = m for all i ∈ N then ( N, A, u ) iscalled an m - strategy game. A game ( N, A, u ) is finite when both N is finite and A i is finite for all i ∈ N .A game may be displayed pictorially as a list of matrices. We list the strategies from players n − n along the rows and columns respectively, and for games with more than two players have a separatematrix for each strategy combination of the remaining players. Each strategy profile s ∈ A correspondsto a unique cell in one of the matrices where the payoffs are written in the form (cid:0) u i ( s ) (cid:1) i ∈ N . a ba , , , , b , , , , a, , ) a ba , , , , b , , , , b, , )We find the payoff to player 3 for the strategy profile ( b, b, a ) ∈ A as follows: reading the strategy profilefrom left to right, player 1 has chosen the second matrix, player 2 has chosen the second row and player3 has chosen the first column, the third value of which is the payoff to player 3. Hence u ( b, b, a ) = 4.A pure strategy Nash equilibrium is a strategy profile s ∈ A where for each i ∈ N , u i ( s i , s − i ) ≥ u i ( s ′ i , s − i )for all s ′ i ∈ A i . For example, in Example 2.1 the profile ( b, b, b ) is a pure strategy Nash equilibrium.We denote the subgroup relation as ≤ , the group generated by a subset H of a group G as h H i , the groupof permutations on a non-empty set N as S N , and the subset of transpositions on N as T N . The readeris reminded that the permutations on N are equivalent to the bijections from N to itself, henceforth wewill refer to them interchangeably.An action of a group G on a set N is a homomorphism α from G into the bijections from N to itself.For each g ∈ G and i ∈ N we denote (cid:0) α ( g ) (cid:1) ( i ) as g ( i ). When G acts on the left or right of N theaction is called a left or right action respectively. We note that left actions can be defined equivalentlyas antihomomorphisms that act on the right, and dually for right actions.An action is transitive if for each i, j ∈ N there exists g ∈ G such that g ( i ) = j , regular if for each i, j ∈ N there exists precisely one g ∈ G such that g ( i ) = j , and n -transitive if for each π ∈ S N thereexists g ∈ G such that g ( i ) = π ( i ) for all i ∈ N . When an action of G can be inferred we simply refer to G as being transitive, regular or n -transitive respectively.The stabiliser of i ∈ N , which we denote as G i , is the subgroup { g ∈ G : g ( i ) = i } of elements in G that fix i . Similarly the stabiliser of N , which we denote as G N , is the normal subgroup { g ∈ G : g ( i ) = i for all i ∈ N } = ∩ i ∈ N G i of elements in G that fix each i ∈ N .The orbit of i ∈ N is G ( i ) = { g ( i ) : g ∈ G } . The orbits of N , denoted as N/G , is the set { G ( i ) : i ∈ N } which forms a partition of N . 2y a groupoid we mean a category in which every morphism is bijective. For the sake of brevity, whenthe objects of a groupoid can be inferred we refer to the morphisms as a groupoid. There are various ways to define a notion of symmetry, not all of which are distinct. In each case weneed all players to have the same number of strategies, consequently all games are implicitly m -strategygames. It is often assumed when defining symmetric games that all players have the same strategylabels and any notion of symmetry will treat the same labels as equivalent. We shall refer to these as label-dependent notions. There is some confusion (Dasgupta and Maskin, 1986) over how to correctly define symmetric games,in order to provide clarity we need to review two ways that player permutations may act on strategyprofiles.Given a permutation π ∈ S N , two action choices are ( s i ) i ∈ N ( s π ( i ) ) i ∈ N and ( s i ) i ∈ N ( s π − ( i ) ) i ∈ N .We denote ( s π − ( i ) ) i ∈ N as π ( s ), for example given ( s , ..., s n ) ∈ A , π ( s , ..., s n ) = ( s π − (1) , ..., s π − ( n ) ).The author notes that our somewhat unintuitive notation has been chosen so that it matches withcomposition and inversion in an ideal manner. That is so for each s ∈ A , ( τ ◦ π )( s ) = τ (cid:0) π ( s ) (cid:1) and( τ ◦ π ) − = π − ◦ τ − . s π ( s ) is a left action of S N on A . Proof.
The identity permutation trivially acts as an identity so we need only establish associativity.For each π, τ ∈ S N , s ∈ A and i ∈ N , (cid:0) ( τ ◦ π )( s ) (cid:1) i = s ( τ ◦ π ) − ( i ) = s π − ( τ − ( i )) = (cid:0) π ( s ) (cid:1) τ − ( i ) = (cid:16) τ (cid:0) π ( s ) (cid:1)(cid:17) i .Since π − ( s ) = ( s π ( i ) ) i ∈ N for all s ∈ A , s π ( s ) and s π − ( s ) are dual to each other. Hence the dualresults hold for π − . s π − ( s ) is a right action of S N on A .Given π ∈ S N we denote the map s u π ( i ) (cid:0) π ( s ) (cid:1) as u π ( i ) ◦ π . For each π, τ ∈ S N , u ( τ ◦ π )( i ) ◦ ( τ ◦ π ) = ( u τ ( π ( i )) ◦ τ ) ◦ π . Proof.
For each i ∈ N , s ∈ A , (cid:0) u ( τ ◦ π )( i ) ◦ ( τ ◦ π ) (cid:1) ( s ) = u ( τ ◦ π )( i ) (cid:0) ( τ ◦ π )( s ) (cid:1) = u τ ( π ( i )) (cid:16) τ (cid:0) π ( s ) (cid:1)(cid:17) = (cid:0) ( u τ ( π ( i )) ◦ τ ) ◦ π (cid:1) ( s ). Game invariants give us a notion of players being indifferent between the current positions and analternative arrangement of positions. π ∈ S N is an invariant of Γ if for each i ∈ N , u i = u π ( i ) ◦ π . The invariants of a game form a group.
Proof.
Since the identity permutation e ∈ S N acts as an identity on A it follows that u i = u i ◦ e for all i ∈ N , hence e is an invariant. Suppose π ∈ S N is an invariant of Γ, and hence that for each i ∈ N , u π − ( i ) = u i ◦ π . Then for each i ∈ N , u i = ( u i ◦ π ) ◦ π − = u π − ( i ) ◦ π − . Finally suppose π, τ ∈ S N areinvariants of Γ. Then for each i ∈ N , u i = u π ( i ) ◦ π = ( u τ ( π ( i )) ◦ τ ) ◦ π = u ( τ ◦ π )( i ) ◦ ( τ ◦ π ).3 .3 Notions of Anonymity Before surveying label-dependent notions of fairness we review various notions of anonymity introducedby Brandt et al. (2009).Central to anonymity is the notion that players do not distinguish between their opponents, by whichwe mean each player merely cares about the strategies being played by their opponents and is indifferentbetween who is playing them.
Γ is weakly anonymous if for each i ∈ N , π ∈ S N −{ i } , u i = u i ◦ π .. Weakly Anonymous 3-player game. a ba , , , , b , , , , a, , ) a ba , , , , b , ,
13 15 , , b, , )The reader may like to verify that u i = u i ◦ ( jk ) for all distinct i, j, k ∈ N . For example, u ( a, b, a ) =4 = u (cid:0) (23)( a, b, a ) (cid:1) = u ( a, a, b ). Since S N −{ i } = { e, ( jk ) } for all i ∈ N , Γ is weakly anonymous.When we say that players do not distinguish between their opponents, we mean for example that whenplaying a , player 1 is indifferent between the strategy profiles ( a, a, b ) and ( a, b, a ).Note we are analysing symmetry under the theme of fairness rather than anonymity. Consequently weare referring to what Brandt et al. (2009) called weakly symmetric and weakly anonymous as weaklyanonymous and anonymous respectively.Weak anonymity may be strengthened by requiring the players care merely about the strategies beingplayed and be indifferent between who is playing each strategy, or equivalently, by requiring each playerhave the same payoff for each orbit in A/S N . Γ is anonymous if for each i ∈ N , π ∈ S N , u i = u i ◦ π . Anonymous 3-player game. a ba , , , , b , , , , a, , ) a ba , , , , b , , , , b, , )The reader may like to verify the orbits of A are given by A/S N = (cid:8) { ( a, a, a ) } , { ( a, a, b ) , ( a, b, a ) , ( b, a, a ) } , { ( a, b, b ) , ( b, a, b ) , ( b, b, a ) } , { ( b, b, b ) } (cid:9) and that each player has the same payofffor each orbit in A/S N .For example, let π = (123), then we have π ( s , s , s ) = ( s π − (1) , s π − (2) , s π − (3) ) = ( s , s , s ) giving us π ( a, a, b ) = ( b, a, a ).Anonymity may be strengthened also by requiring all players have the same payoff for each orbit in A/S N . Γ is fully anonymous if for each i, j ∈ N , π ∈ S N , u i = u j ◦ π . Fully anonymous 3-player game. a ba , , , , b , , , , a, , ) a ba , , , , b , , , , b, , )The orbits of A for the above game are the same as in Example 3.9, however now all players have thesame payoff for each orbit.In a fully anonymous game each player is indifferent between which position they play. Hence fullyanonymous games are one class of games that fall under our broad requirements for fairness.4 .4 Notions of Fairness Our broad requirements for fairness that players be indifferent between which position they play may bemade more precise by requiring the invariants of a game be a transitive subgroup of S N . Γ is standard symmetric (Stein, 2011) if there exists a transitive subgroup H of theplayer permutations such that for each i ∈ N and π ∈ H , u i = u π ( i ) ◦ π .In a standard symmetric game, while being indifferent between which position they play, each player maycare about the arrangement of their opponents, or alternatively may distinguish between their opponents. Standard symmetric 3-player game. a ba , , , , b , , , , a, , ) a ba , , , , b , , , , b, , )The reader may like to verify that Γ is invariant under (123) and not invariant under (12). Since h (123) i = { e, (123) , (132) } is a transitive subgroup of S , Γ is standard symmetric. Furthermore since(12) is not an invariant the players are not indifferent between all possible position arrangements.A useful analogy for considering the fairness of Γ is a game with three players sitting in a circle such thateach player is indifferent between circular rotations of positions, and not indifferent to their opponentsswapping positions.We obtain a stronger level of fairness by requiring the players be indifferent between all possible positionrearrangements, that is by requiring all player permutations be invariants. Γ is fully symmetric if it is invariant under S N .The reader may like to verify that Example 2.1 is invariant under the permutations (12) and (123). Forexample, let π = (123), then π ( s , s , s ) = ( s , s , s ) giving us u ( b, a, a ) = u ( a, b, a ) = u ( a, a, b ) = 3.Since invariants are closed under composition and h (12) , (123) i = S , Example 2.1 is fully symmetric.Next we establish that Definition 3.14 can be characterised by various conditions. The following conditions are equivalent:(i) Γ is fully symmetric;(ii) Γ is standard symmetric and weakly anonymous;(iii) For each i ∈ N and π ∈ S N , u π ( i ) = u i ◦ π − ;(iv) For each i ∈ N and τ ∈ T N , u i = u τ ( i ) ◦ τ ; and(v) For each i ∈ N and τ ∈ T N , u i = u τ ( i ) ◦ τ − . Proof.
Condition (ii) follows trivially from Condition (i). Now suppose Condition (ii) is satisfied and let H be a transitive subgroup of player permutations under which Γ is invariant. Let π ∈ S N , i ∈ N and τ ∈ H such that τ ( i ) = π ( i ). Since ( τ − ◦ π ) ∈ S N −{ i } it follows from weak anonymity that u i = u i ◦ ( τ − ◦ π ).It also follows from standard symmetry that u i = u τ ( i ) ◦ τ , putting these two bits of information togetherwe have u i = u i ◦ ( τ − ◦ π ) = ( u τ ( i ) ◦ τ ) ◦ ( τ − ◦ π ) = u τ ( i ) ◦ ( τ ◦ τ − ) ◦ π = u τ ( i ) ◦ π = u π ( i ) ◦ π .Suppose Condition (i) is satisfied, then for each i ∈ N and π ∈ S N , u π ( i ) = u π ( i ) ◦ ( π ◦ π − ) =( u π ( i ) ◦ π ) ◦ π − = u i ◦ π − . The converse works the same in reverse giving equivalence of Conditions (i)and (iii).Condition (i) implies Condition (iv) since T N ⊆ S N , and Condition (iv) implies Condition (i) directlyfrom Corollary 3.3 and that h T N i = S N . Conditions (iv) and (v) are equivalent since each transpositionis its own inverse.Condition (iii) in Theorem 3.15 was used by von Neumann and Morgenstern (1944), which was ideal fortheir chosen notation of permutations acting on the right of players and strategy profiles. Of course anygenerating set of S N may replace T N in Condition (iv) of Theorem 3.15.5otably it is easy to mistakingly use the following inequivalent condition: for each i ∈ N and π ∈ S N , u i = u π ( i ) ◦ π − (Dasgupta and Maskin, 1986). However this does not permute the players and strategyprofiles correctly as the right hand side does not have player π ( i ) playing the strategy that player i isplaying, which we illustrate using Example 2.1.Let π = (123) ∈ S , the incorrect condition given by Dasgupta and Maskin (1986) requires that foreach i ∈ N and ( s , s , s ) ∈ A , we have u i ( s , s , s ) = u π ( i ) ( s π (1) , s π (2) , s π (3) ) = u π ( i ) ( s , s , s ). Byconsidering ( b, a, a ) ∈ A , we see that 3 = u ( b, a, a ) = u ( a, a, b ) = 2. It should be fairly obvious that ifwe are mapping player 1 to player 2 and player 1 is playing b then we want the mapped strategy profileto have player 2 playing b .Since T N ⊆ S N , it follows from Condition (v) in Theorem 3.15 that the incorrect condition fromDasgupta and Maskin (1986) is somewhat surprisingly a more restrictive condition than the conditionsin Theorem 3.15. When n = 2, since each transposition is its own inverse, the incorrect condition fromDasgupta and Maskin (1986) is equivalent to the conditions in Theorem 3.15. We now establish thatfor n ≥ (Brandt et al., 2009) The following conditions are equivalent:(i) Γ is fully anonymous; and(ii) Γ is fully symmetric and u i = u j for all i, j ∈ N . Let π, τ ∈ S N . If u i = u π ( i ) ◦ π − = u τ ( i ) ◦ τ − for all i ∈ N then u i = u ( τ ◦ π )( i ) ◦ ( π ◦ τ ) − for all i ∈ N . Proof.
For each i ∈ N , u i = u π ( i ) ◦ π − = ( u τ ( π ( i )) ◦ τ − ) ◦ π − = u ( τ ◦ π )( i ) ◦ ( π ◦ τ ) − . If n ≥ u i = u j for all i, j ∈ N ; and(ii) For each i ∈ N and π ∈ S N , u i = u π ( i ) ◦ π − . Proof.
Suppose Condition (i) holds, then for each i ∈ N and π ∈ S N , u i = u π − ( i ) ◦ π − = u π ( i ) ◦ π − .Conversely suppose Condition (ii) holds, and hence that Γ is fully symmetric. Let i, j, k ∈ N be distinct.Since ( ik ) ◦ ( ijk ) ◦ ( jk ) = ( ijk ) and (cid:0) ( jk ) ◦ ( ijk ) ◦ ( ik ) (cid:1) − = ( ik ) ◦ ( ikj ) ◦ ( jk ) = e , it follows from Lemma3.17 that u i = u j . There are two important reasons why our simplifying assumption that players have the same strategylabels leaves our analysis incomplete. Our first reason is that relabelling the strategies for a standardsymmetric game leads to a strategically equivalent game that may no longer be considered symmetricinside our label-dependent framework.Ideally we want to be able to determine when two games merely differ by player and strategy labelswithout having to go through and check all possible rearrangements of the labels.Our second reason is that there are weaker notions of fairness that cannot be captured within our label-dependent framework. As a motivating example consider Matching Pennies.
Matching Pennies
H TH , − − , T − , , − .1 Game Bijections A game bijection from Γ = ( N, A, u ) to Γ = ( M, B, v ) consists of a bijection π : N → M and for each player i ∈ N , a bijection τ i : A i → B π ( i ) , which we denote as (cid:0) π ; ( τ i ) i ∈ N (cid:1) .We denote the set of game bijections from Γ to Γ as Bij(Γ , Γ ), or simply S Γ for the bijections from agame Γ to itself. Let g = (cid:0) π ; ( τ i ) i ∈ N (cid:1) ∈ Bij(Γ , Γ ), i ∈ N , s i ∈ A i and s ∈ A , using similar notation toour label-dependent framework we denote π ( i ) as g ( i ), τ i ( s i ) as g ( s i ), (cid:0) τ π − ( j ) ( s π − ( j ) ) (cid:1) j ∈ M ∈ B as g ( s )giving (cid:0) g ( s ) (cid:1) g ( i ) = τ i ( s i ) = g ( s i ), and the map s u g ( i ) (cid:0) g ( s ) (cid:1) as u g ( i ) ◦ g . Consider the following 2-player games. c da , , b , , g he , , f , , Given ( a, c ) ∈ A and g = (cid:0) (12); (cid:0) a bg h (cid:1) , (cid:0) c df e (cid:1)(cid:1) ∈ Bij(Γ , Γ ), g ( a, c ) = ( f, g ).Let Γ = ( L, C, w ) also be a game. For g = (cid:0) π ; ( τ i ) i ∈ N (cid:1) ∈ Bij(Γ , Γ ) and h = (cid:0) η ; ( φ j ) j ∈ M (cid:1) ∈ Bij(Γ , Γ ),their composite , denoted h ◦ g , is (cid:0) η ◦ π ; ( φ π ( i ) ◦ τ i ) i ∈ N (cid:1) ∈ Bij(Γ , Γ ), and the inverse of g , denoted g − ,is (cid:0) π − ; ( τ − π − ( j ) ) j ∈ M (cid:1) ∈ Bij(Γ , Γ ). Consider Example 3.13 except with strategy labels A = { c, d } and A = { e, f } forplayers 2 and 3 respectively. We compose and invert bijections g = (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fb a (cid:1)(cid:1) , h = (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1) , (cid:0) e ff e (cid:1)(cid:1) ∈ S Γ as follows: h ◦ g = (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1) , (cid:0) e ff e (cid:1)(cid:1) ◦ (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fb a (cid:1)(cid:1) = (cid:0) (12) ◦ (123); (cid:0) c da b (cid:1) ◦ (cid:0) a bd c (cid:1) , (cid:0) e ff e (cid:1) ◦ (cid:0) c de f (cid:1) , (cid:0) a bc d (cid:1) ◦ (cid:0) e fb a (cid:1)(cid:1) = (cid:0) (23); (cid:0) a bb a (cid:1) , (cid:0) c df e (cid:1) , (cid:0) e fd c (cid:1)(cid:1) ; and g − = (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fb a (cid:1)(cid:1) − = (cid:0) (123) − ; (cid:0) e fb a (cid:1) − , (cid:0) a bd c (cid:1) − , (cid:0) c de f (cid:1) − (cid:1) = (cid:0) (132); (cid:0) a bf e (cid:1) , (cid:0) c db a (cid:1) , (cid:0) e fc d (cid:1)(cid:1) . ( h ◦ g )( s ) = h ( g ( s )) for all s ∈ A . Proof. ( h ◦ g )( s ) = (cid:0) η ◦ π ; ( φ π ( i ) ◦ τ i ) i ∈ N (cid:1) ( s )= (cid:0) φ η − ( k ) ◦ τ ( η ◦ π ) − ( k ) ( s ( η ◦ π ) − ( k ) ) (cid:1) k ∈ L = (cid:16) φ η − ( k ) (cid:0) τ π − ( η − ( k )) ( s π − ( η − ( k )) ) (cid:1)(cid:17) k ∈ L = (cid:16) φ η − ( k ) (cid:0) g ( s ) η − ( k ) (cid:1)(cid:17) k ∈ L = (cid:16) h (cid:0) g ( s ) (cid:1) k (cid:17) k ∈ L = h ( g ( s )) . u ( h ◦ g )( i ) ◦ ( h ◦ g ) = ( u h ( g ( i )) ◦ h ) ◦ g for all i ∈ N . Proof.
This follows identically to the proof of Corollary 3.3.
Game bijections form a groupoid.
Proof.
Let Γ = ( P, C ), Γ = ( Q, D ), f = (cid:0) π ; ( τ i ) i ∈ N (cid:1) ∈ Bij(Γ , Γ ), g = (cid:0) η ; ( φ j ) j ∈ M (cid:1) ∈ Bij(Γ , Γ ), h = (cid:0) ξ ; ( λ k ) k ∈ P (cid:1) ∈ Bij(Γ , Γ ). Then: f ◦ id Γ = (cid:0) π ◦ id N ; ( τ i ◦ id A i ) i ∈ N (cid:1) f = (cid:0) id M ◦ π ; (id B π ( i ) ◦ τ i ) i ∈ N (cid:1) = id Γ ◦ f ; f ◦ f − = (cid:0) π ◦ π − ; ( τ π − ( j ) ◦ τ − π − ( j ) ) j ∈ M (cid:1) = id Γ ; f − ◦ f = (cid:0) π − ◦ π ; ( τ − π − ( π ( i )) ◦ τ i ) i ∈ N (cid:1) = id Γ ; and h ◦ ( g ◦ f ) = (cid:0) ξ ; ( λ k ) k ∈ P (cid:1) ◦ (cid:0) η ◦ π ; ( φ π ( i ) ◦ τ i ) i ∈ N (cid:1) = (cid:0) ξ ◦ η ◦ π ; ( λ ( η ◦ π )( i ) ◦ φ π ( i ) ◦ τ i ) i ∈ N (cid:1) = (cid:0) ξ ◦ η ; ( λ η ( j ) ◦ φ j ) j ∈ M (cid:1) ◦ (cid:0) π ; ( τ i ) i ∈ N (cid:1) = ( h ◦ g ) ◦ f. Game isomorphisms are game bijections that preserve strategic structure, they are useful for establishingstrategic equivalence between games, or as we will be using them, for considering label-independentnotions of symmetry.We will only require the strictest notion of game isomorphism to explore label-independent notions ofsymmetry, treating two games as isomorphic when they differ only by the player and strategy labels.However one can define ordinal and cardinal game isomorphisms by requiring preservation of preferencesover pure and mixed strategy profiles respectively, then characterise each by the existence of increasingmonotonic and affine transformations respectively (Harsanyi and Selten, 1988).
A bijection g ∈ Bij(Γ , Γ ) is a game isomorphism if u i = v g ( i ) ◦ g for all i ∈ N .We denote by Isom(Γ , Γ ) the set of isomorphisms from Γ to Γ . The reader may like to verify thatthe bijection in Example 4.3 is in fact an isomorphism. For example, u ( a, d ) = v g (1) (cid:0) g ( a, d ) (cid:1) = v ( e, g ). Game isomorphisms form a groupoid.
Proof.
For each g ∈ Isom(Γ , Γ ) and j ∈ M , v j = ( v j ◦ g ) ◦ g − = u g − ( j ) ◦ g − , giving us g − ∈ Isom(Γ , Γ ). Let Γ = ( P, C, w ), then for each g ∈ Isom(Γ , Γ ), h ∈ Isom(Γ , Γ ) and i ∈ N , u i = v g ( i ) ◦ g = ( w h ( g ( i )) ◦ h ) ◦ g = w ( h ◦ g )( i ) ◦ ( h ◦ g ), giving us ( h ◦ g ) ∈ Isom(Γ , Γ ).The remaining conditions follow from Theorem 4.7. If Γ ∼ = Γ ∼ = Γ then Isom(Γ , Γ ) ∼ = Isom(Γ , Γ ).Game isomorphisms induce an equivalence relation where games in the same equivalence class have thesame strategic structure. There is a finite number of ordinal equivalence classes for games with both afixed number of players and fixed number of strategies for each of the players. Goforth and Robinson(2005) counted 144 ordinal equivalence classes for the 2-player 2-strategy games. The bijections S Γ from a game to itself form a group that acts on the players and strategy profiles. Infact for an m -strategy game S Γ is isomorphic to the wreath product S N ≀ S M where M = { , ..., m } ,which may be seen by setting A i = M for all i ∈ N .Let G be a subgroup of S Γ . We denote the subgroup of player permutations used by G as −→ G . Furthermore,we say that G is player transitive if G acts transitively on N , player n -transitive if G acts n -transitivelyon N , and only-transitive if G acts transitively and not n -transitively on N . Two bijections g, h ∈ G have the same player permutation if and only if they are in thesame coset of G/G N . Proof.
Suppose g, h have the same player permutation, then h = g ◦ ( g − ◦ h ) ∈ ( g ◦ G N ). The converseis obvious.Hence the factor group G/G N merely tells us what player permutations are used by G . G/G N ∼ = −→ G . 8he isomorphisms from a game to itself form a subgroup of the game bijections called the automorphismgroup of Γ, which we denote as Aut(Γ). Game automorphisms capture the notion of players beingindifferent between the current positions and an alternative arrangement of positions. Note our definitionis equivalent to the definition used by Nash (1951).For the sake of brevity, we refer to a subgroup of Aut(Γ) as a subgroup of Γ, denote the stabilisersubgroup of Aut(Γ) on N as Γ N , and denote the player permutations used by Aut(Γ) as −→ Γ .
Now that players need not have the same strategy labels, we seek a way to determine which subgroupsof S Γ act on strategy profiles in an equivalent way to permutations for some relabelling of the strategies.Stein (2011) introduced strategy triviality for this purpose. A subgroup G of S Γ is strategy trivial (Stein, 2011) if for each i ∈ N , g ( s i ) = s i forall g ∈ G i and s i ∈ A i . (Stein, 2011) If G is strategy trivial then for each g, h ∈ G such that g ( i ) = h ( i ), g ( s i ) = h ( s i ) for all s i ∈ A i . Proof.
Since ( g − ◦ h ) ∈ G i , by strategy triviality, g ( s i ) = g (cid:0) ( g − ◦ h )( s i ) (cid:1) = ( g ◦ g − ) (cid:0) h ( s i ) (cid:1) = h ( s i ). If G is strategy trivial then G N = { id Γ } .Hence strategy trivial subgroups have at most one bijection for each player permutation. Example 5.9establishes that the converse of Corollary 4.15 is false. If G ≤ S Γ is strategy trivial then for each i ∈ N and τ ∈ −→ G , there exists g iτ ( i ) ∈ Bij( A i , A τ ( i ) ) such that G = { ( π ; ( g iπ ( i ) ) i ∈ N ) : π ∈ −→ G } .It follows that all paths from one player to another map the strategies in a canonical manner. Hence if G is also player transitive then the strategy sets are matched such that they can be treated as the sameset. We now introduce matchings to formalise what is meant by the strategy sets being matched. A matching of A , ..., A n is a relation M ⊆ × i ∈ N A i which is i -total and i -unique forall i ∈ N . Let A = { a, b } , A = { c, d } and A = { e, f } . One matching of A × A × A is M = { ( a, d, f ) , ( b, c, e ) } . a c eb d f From a game theoretic point of view, a matching is a subset M of the strategy profiles where for each i ∈ N and a i ∈ A i there is exactly one s ∈ M such that s i = a i , and hence | M | = m .For each i, j ∈ N , a matching M induces a bijection M ij ∈ Bij( A i , A j ) where, given a i ∈ A i , M ij ( a i )is the unique a j ∈ A j such that there exists s ∈ M with s i = a i and s j = a j . For example given thematching in Example 4.18, M = (cid:0) e fb a (cid:1) . { M ij : i, j ∈ N } is a groupoid. Proof.
It follows by definition that for each i, j, k ∈ N , M ii = id A i , M − ij = M ji and M jk ◦ M ij = M ik .Now for each i, j, k, l ∈ N , M ij ◦ M ii = M ij = M jj ◦ M ij , M kl ◦ ( M jk ◦ M ij ) = M kl ◦ M ik = M il = M jl ◦ M ij = ( M kl ◦ M jk ) ◦ M ij , M ij ◦ M − ij = M ij ◦ M ji = M jj and M − ij ◦ M ij = M ji ◦ M ij = M ii .Furthermore, for each π ∈ S N , a matching M induces a game bijection (cid:0) π ; ( M iπ ( i ) ) i ∈ N (cid:1) ∈ S Γ , which wedenote as M π . For example given the matching in Example 4.18, M (13) = (cid:0) (13); (cid:0) a bf e (cid:1) , (cid:0) c dc d (cid:1) , (cid:0) e fb a (cid:1)(cid:1) .For each G ⊆ S N we denote the set { M π : π ∈ G } of bijections induced by G as M G . M : S N → S Γ is a homomorphism. Proof.
Let π, φ ∈ S N , then M φ ◦ M π = (cid:0) φ ; ( M iφ ( i ) ) i ∈ N (cid:1) ◦ (cid:0) π ; ( M iπ ( i ) ) i ∈ N (cid:1) = (cid:0) φ ◦ π ; ( M π ( i )( φ ◦ π )( i ) ◦ M iπ ( i ) ) i ∈ N (cid:1) = (cid:0) φ ◦ π ; ( M i ( φ ◦ π )( i ) ) i ∈ N (cid:1) = M ( φ ◦ π ) . 9 .21 Corollary: M π − = M − π for all π ∈ S N . For each π ∈ S N , M π ( s ) = s for all s ∈ M . Proof.
For each i ∈ N , (cid:0) M π ( s ) (cid:1) i = M π − ( i ) i ( s π − ( i ) ) = s i .If we relabel the strategies played in each s ∈ M to be the same, giving players the same strategy labels,then each permutation π ∈ S N acts on our relabelled strategy profiles equivalently to how M π actson our original strategy profiles. Hence a subgroup G of S Γ acts on strategy profiles equivalently topermutations for some relabelling of the strategies precisely when G = M −→ G for some matching M , whichwe now establish occurs precisely when G is strategy trivial. Let G ≤ S Γ be player transitive. There exists a matching M such that M −→ G = G if andonly if G is strategy trivial. Proof.
Suppose there exists a matching M such that M −→ G = G . That M −→ G ≤ S Γ follows from Lemma4.20. Now for each i ∈ N and g ∈ G i , M ig ( i ) = M ii = id A i .Conversely suppose G is strategy trivial. By Corollary 4.16, for each i ∈ N and τ ∈ −→ G there exists g iτ ( i ) ∈ Bij( A i , A τ ( i ) ) such that G = { (cid:0) π ; ( g iπ ( i ) ) i ∈ N (cid:1) : π ∈ −→ G } .Let i ∈ N and M = { ( g ij ( a i )) j ∈ N : a i ∈ A i } . M is a matching since for each j ∈ N and a j ∈ A j , thereexists a unique strategy a i ∈ A i for player i such that g ij ( a i ) = a j . Furthermore M is independent of i since for each k ∈ N , (cid:0) g ij ( a i ) (cid:1) j ∈ N = (cid:0) ( g kj ◦ g ik )( a i ) (cid:1) j ∈ N . Hence M kl = g kl for all k, l ∈ N , giving us M π = (cid:0) π ; ( M iπ ( i ) ) i ∈ N (cid:1) = (cid:0) π ; ( g iπ ( i ) ) i ∈ N (cid:1) ∈ G for all π ∈ −→ G .Hence weakly anonymous games may be characterised as follows, similarly for anonymous and fullyanonymous games. The following conditions are equivalent:(i) There exists weakly anonymous Γ ′ such that Γ ∼ = Γ ′ ;(ii) There exists player n -transitive and strategy trivial G ≤ Γ such that for each i ∈ N and g ∈ G i , u i = u i ◦ g ; and(iii) There exists a matching M such that for each i ∈ N and π ∈ S N −{ i } , u i = u i ◦ M π .We denote by M ( n, m ) the set of matchings for an n -player m -strategy game. (i) If m = n = 2 then, letting A = { a, b } and A = { c, d } , M (2 ,
2) = (cid:8) { ( a, c ) , ( b, d ) } , { ( a, d ) , ( b, c ) } (cid:9) . (ii) If m = 3 and n = 2 then, letting A = { a, b, c } and A = { d, e, f } , M (2 ,
3) = (cid:8) { ( a, d ) , ( b, e ) , ( c, f ) } , { ( a, d ) , ( b, f ) , ( c, e ) } , { ( a, e ) , ( b, d ) , ( c, f ) } , { ( a, e ) , ( b, f ) , ( c, d ) } , { ( a, f ) , ( b, d ) , ( c, e ) } , { ( a, f ) , ( b, e ) , ( c, d ) } (cid:9) . There are a number of ways to count the number of matchings in M ( n, m ). Below we present one, thoughnote an alternative is to establish that M ( n, m ) ∼ = Bij( A , A ) × ... × Bij( A n − , A n ). For each n ≥ M ( n,
2) is a partition of A ; and | M ( n, | = 2 n − . Proof.
For each s ∈ A , the profile s ′ where each player swaps their strategy choice is the unique profilein A such that { s, s ′ } ∈ M ( n, | M ( n, | = | A | = 2 n − . For each n ≥ m ≥ | M ( n, m ) | = m n − | M ( n, m − | . Proof.
Let i ∈ N . Each a i can be matched with each a − i ∈ A − i and | A − i | = m n − . Furthermore, foreach ( a i , a − i ) there are | M ( n, m − | ways to match the remaining m − n players. For each m, n ≥ | M ( n, m ) | = ( m !) n − . Proof.
This follows inductively from Lemmas 4.26 and 4.27.10
Label-Independent Notions of Symmetry
Similar to our label-independent characterisations of our label-dependent notions of anonymity, Theorem4.23 gives us the following label-independent characterisations of our label-dependent notions of fairness.
The following conditions are equivalent:(i) There exists standard symmetric Γ ′ such that Γ ∼ = Γ ′ ;(ii) Γ has a player transitive and strategy trivial subgroup G ; and(iii) There exists a matching M and player transitive T ≤ S N such that M T ≤ Aut(Γ).
The following conditions are equivalent:(i) There exists fully symmetric Γ ′ such that Γ ∼ = Γ ′ ;(ii) Γ has a player n -transitive and strategy trivial subgroup G ; and(iii) There exists a matching M such that M S N ≤ Aut(Γ).Henceforth we will use fully and standard symmetric to refer to our label-independent characterisations.
If Γ is standard symmetric then there exists a matching M such that for each s ∈ M , u i ( s ) = u j ( s ) for all i, j ∈ N . Proof.
This follows from Lemma 4.22.Remember that the defining features for standard and fully symmetric games inside our label-dependentframework were that players be indifferent between which position they play and the arrangement ofthe players respectively. Inside our label-independent framework, these defining features capture largerclasses of fair games.
A game is symmetric (Stein, 2011) if its automorphism group is player transitive and n -transitively symmetric if its automorphism group is player n -transitive. The automorphism group of Matching Pennies in Example 4.1 isAut(Γ) = h (cid:0) (12); (cid:0) H TH T (cid:1) , (cid:0) H TT H (cid:1)(cid:1) i = { (cid:0) e ; (cid:0) H TH T (cid:1) , (cid:0) H TH T (cid:1)(cid:1) , (cid:0) e ; (cid:0) H TT H (cid:1) , (cid:0) H TT H (cid:1)(cid:1) , (cid:0) (12); (cid:0) H TH T (cid:1) , (cid:0) H TT H (cid:1)(cid:1) , (cid:0) (12); (cid:0) H TT H (cid:1) , (cid:0) H TH T (cid:1)(cid:1) } . Since Aut(Γ) is player transitive, is not strategy trivial and contains no proper transitive subgroups,Matching Pennies is an n -transitively non-standard symmetric game.Peleg et al. (1999), Sudh¨olter et al. (2000) considered a game symmetric if Aut(Γ) / Γ N ∼ = S N . It followsimmediately from Corollary 4.12 that this is equivalent to a game being n -transitively symmetric, andfurthermore that Aut(Γ) / Γ N being isomorphic to some transitive subgroup of S N is equivalent to a gamebeing symmetric.We now consider games which have a subgroup G isomorphic to S N with G N = { id Γ } . Fully symmetricgames obviously satisfy this condition, Example 5.9 shows that the converse of this is false. Below weshow that all games satisfying this condition are n -transitively standard symmetric games; the authorhas been unable to show whether the converse holds. If Γ has a subgroup G isomorphic to S N with G N = { id Γ } then it is n -transitivelystandard symmetric. Proof. n -transitivity of Γ follows from −→ G = S N . Now since each n -cycle generates a regular subgroupof S N , the subgroup of G generated by an automorphism whose player permutation is an n -cycle istransitive and strategy trivial, hence Γ is standard symmetric.11e end our exploration of symmetry notions with games that have a transitive subgroup G isomorphicto −→ Γ with G N = { id Γ } . Standard symmetric games obviously satisfy this condition. To look at theconverse we consider the argument used in Proposition 5.6.If all transitive subgroups of S N had regular subgroups then games with a transitive subgroup G iso-morphic to −→ Γ with G N = { id Γ } would be standard symmetric. However this is not the case, Hulpke(2005) listed the non-regular minimally transitive permutation subgroups up to degree 30. The smallestexample is h (14) ◦ (25) , (135) ◦ (246) i of degree 6 and order 12.We will see in Example 5.13 that games which have a transitive subgroup G isomorphic to −→ Γ with G N = { id Γ } need not be standard symmetric. While our distinct symmetry notions give us various descriptive definitions of strategic fairness, theydo not give us a constructive way to determine where a particular game lies. We now discuss variousstrategies for classifying a game. A discussion on finding automorphisms of games can be found inGabarr´o et al. (2007).To test whether a game Γ is fully or standard symmetric: we first try to construct a matching M ofthe strategy sets where for each profile s ∈ M , all players have the same payoff. If no such matchingexists Γ is neither fully nor standard symmetric. For example in Matching Pennies, since there does notexist a strategy profile where all players receive the same payoff, we can conclude Matching Pennies isnon-standard symmetric.If such matchings exist: to test for full symmetry we check whether such a matching induces automor-phisms for permutations that generate S N ; and to test for standard symmetry we check whether sucha matching induces automorphisms for player permutations that generate a transitive subgroup of S N ,noting that to conclude non-standard symmetry we must check that the game is not invariant under thebijections induced by any such matching and transitive subgroup of S N .The reader should note that every n -cycle generates a transitive subgroup of S N , but not all transitivesubgroups of S N contain an n -cycle. For example the Klein group { e, (12) ◦ (34) , (13) ◦ (24) , (14) ◦ (23) } is a transitive subgroup of S that does not contain any 4-cycles.To test for n -transitivity we check whether there exists automorphisms for permutations that generate S N ;and to test for symmetry (ie. transitivity) we check whether there exists automorphisms for permutationsthat generate a transitive subgroup of S N , again noting that to conclude that a game is not symmetricwe must check that the game is not invariant under any transitive subgroup of S Γ .If we know a game is symmetric (ie. transitive) and want to show it is only-transitive, a sufficientcondition is to find a strategy profile s ∈ A whose payoffs do not appear elsewhere under all possiblepermutations. For example consider Example 3.13 and suppose it has an automorphism whose playerpermutation is (23). The payoffs for the profile ( a, a, b ) are (3 , , s ∈ A with payoffs (3 , , Given a subset G of game bijections we construct the parameterised game Γ( G ) of G as follows: for each g ∈ h G i , s ∈ A and i ∈ N , set u i ( s ) = u g ( i ) (cid:0) g ( s ) (cid:1) . Since automorphisms are closed under compositionwe have h G i ≤ Aut(Γ), hence each orbit of ( N × A ) / h G i has the same payoff. Let G = { (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1)(cid:1) } . For Γ( G ) we require: u ( a, c ) = u ( a, c ) = α u ( a, d ) = u ( b, c ) = γu ( b, c ) = u ( a, d ) = β u ( b, d ) = u ( b, d ) = δc da α, α γ, βb β, γ δ, δ Γ( G )12e call α, β, γ, δ ∈ R the parameters of Γ( G ). Note that distinct parameter choices may lead to strate-gically inequivalent games, even though both games will have the same automorphism group. All fullysymmetric 2-player 2-strategy games are isomorphic to Γ( G ) for at least one choice of parameters, henceΓ( G ) is a general form for fully symmetric 2-player 2-strategy games, or equivalently standard symmetric2-player 2-strategy games.We can define a partial order ≤ on parameterised games as follows: Γ( G ) ≤ Γ( G ′ ) when given a set ofparameter choices for Γ( G ′ ) there exists a set of parameter choices for Γ( G ) such that Γ( G ) ∼ = Γ( G ′ ). Weillustrate our order in Examples 5.8 and 5.10 using the Hasse diagrams for ≤ on parameterised symmetric2-player and 3-player 2-strategy games up to isomorphism. Hasse diagram for ≤ on parameterised symmetric 2-player 2-strategy games up to iso-morphism. Γ( G ) α, α α, αα, α α, α Γ( G ) α, β β, αβ, α α, β Γ( G ) α, α β, ββ, β α, α Γ( G ) α, α β, γγ, β δ, δG = { (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1)(cid:1) } , G = G ∪ { (cid:0) (12); (cid:0) a bd c (cid:1) , (cid:0) c db a (cid:1)(cid:1) } , G = { (cid:0) (12); (cid:0) a bd c (cid:1) , (cid:0) c da b (cid:1)(cid:1) } , G = G ∪ G .To construct a symmetric game or an n -transitively symmetric game we use bijections that generate aplayer transitive or player n -transitive subgroup respectively.To construct an only-transitive symmetric game it is not sufficient to use bijections that generate anonly-transitive subgroup, we must construct Γ( G ) and check that it is only-transitive. This is due to h G i possibly being a proper subgroup of Aut(Γ). For example, if we take: G = { (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fa b (cid:1)(cid:1) , (cid:0) (123); (cid:0) a bc d (cid:1) , (cid:0) c df e (cid:1) , (cid:0) e fa b (cid:1)(cid:1) } , then N × A has one orbit under h G i (i.e. Γ( G ) has one parameter/payoff) despite h G i being an only-transitive subgroup.To construct a standard symmetric game we use the bijections induced from a matching of the strategysets and player permutations which generate a transitive subgroup of S N . To construct a non-standardsymmetric game, we first choose game bijections which are not obviously from the same matching,construct Γ( G ) and check whether it is non-standard symmetric. We construct fully and non-fullysymmetric games similarly. So far we have seen examples of fully symmetric, only-transitive standard symmetric and n -transitivelynon-standard symmetric games. We now look at examples constructed with the process outlined inSubsection 5.3 to show that our notions of symmetry are related as shown in the following Euler diagram.13ullystandard n -transitivesymmetric n -transitively non-fully standard symmetric 3-player game. e fc α, α, α β, γ, δd γ, δ, β δ, γ, β ( a, , ) e fc δ, β, γ β, δ, γd γ, β, δ α, α, α ( b, , ) G = { (cid:0) (123); (cid:0) a bc d (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fa b (cid:1)(cid:1) , (cid:0) (12); (cid:0) a bd c (cid:1) , (cid:0) c db a (cid:1) , (cid:0) e ff e (cid:1)(cid:1) } Since h G i is n -transitive, and the first generator generates a player transitive and strategy trivial groupwith the matching M = { ( a, c, e ) , ( b, d, f ) } , Γ( G ) is n -transitively and standard symmetric. Furthermoresince the bijections induced by M from player transpositions are not automorphisms, Γ( G ) is non-fullysymmetric. Hasse diagram for ≤ on parameterised symmetric 3-player 2-strategy games up toisomorphism. Γ( G ) α, α, α α, α, αα, α, α α, α, αα, α, α α, α, αα, α, α α, α, α Γ( G ) α, α, α β, β, ββ, β, β α, α, αβ, β, β α, α, αα, α, α β, β, β Γ( G ) α, α, α β, β, δβ, δ, β δ, β, βδ, β, β β, δ, ββ, β, δ α, α, α Γ( G ) α, α, α β, β, δβ, δ, β σ, ρ, ρδ, β, β ρ, σ, ρρ, ρ, σ ω, ω, ω Γ( G ) α, α, α β, γ, δγ, δ, β δ, γ, βδ, β, γ β, δ, γγ, β, δ α, α, α Γ( G ) α, α, α β, γ, δγ, δ, β δ, β, γδ, β, γ γ, δ, ββ, γ, δ α, α, α Γ( G ) α, α, α β, γ, δγ, δ, β σ, ρ, τδ, β, γ τ, σ, ρρ, τ, σ ω, ω, ωG = { (cid:0) (123); (cid:0) a bc d (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fa b (cid:1)(cid:1) } , G = G ∪ { (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1) , (cid:0) e fe f (cid:1)(cid:1) } , G = G ∪ { (cid:0) (12); (cid:0) a bd c (cid:1) , (cid:0) c db a (cid:1) , (cid:0) e ff e (cid:1)(cid:1) } ,14 = { (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c df e (cid:1) , (cid:0) e fb a (cid:1)(cid:1) } , G = G ∪ { (cid:0) (123); (cid:0) a bd c (cid:1) , (cid:0) c df e (cid:1) , (cid:0) e fa b (cid:1)(cid:1) } , G = G i ∪ G j for all distinct i, j ∈ { , , } , G = G ∪ G .Cheng et al. (2004) showed that fully symmetric 2-strategy games have at least one pure strategy Nashequilibrium. They also noted that Rock, Paper, Scissors is an example of a fully symmetric 2-player 3-strategy game with no pure strategy Nash equilibria, and indirectly that Matching Pennies is an exampleof a non-standard symmetric 2-player 2-strategy game which has no pure strategy Nash equilibria. Thereader may like to verify that Example 3.13 is a standard symmetric 2-strategy game with no purestrategy Nash equilibria.Note Example 5.9 is the only parameterised n -transitively non-fully standard symmetric 3-player 2-strategy game up to isomorphism. Furthermore note there are pure strategy Nash equilibria for eachchoice of parameters. The author has been unable to show whether the result from Cheng et al. (2004)weakens to n -transitively standard symmetric 2-strategy games. Two only-transitive non-standard symmetric 4-player games. g he α, β, γ, δ ρ, τ, σ, ωf σ, ω, ρ, τ ω, ρ, τ, σ ( a, c, , ) g he δ, α, β, γ τ, σ, ω, ρf γ, δ, α, β β, γ, δ, α ( a, d, , ) g he β, γ, δ, α γ, δ, α, βf τ, σ, ω, ρ δ, α, β, γ ( b, c, , ) g he ω, ρ, τ, σ σ, ω, ρ, τf ρ, τ, σ, ω α, β, γ, δ ( b, d, , ) G = { (cid:0) (1234); (cid:0) a bd c (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fg h (cid:1) , (cid:0) g ha b (cid:1)(cid:1) } Since there does not exist any profile where the payoffs are equal and h G i is transitive, Γ( G ) is non-standard symmetric. Now for the strategy profile ( a, c, e, g ) we have payoffs ( α, β, γ, δ ). If Γ( G ) had anautomorphism using (23) then there would be a strategy profile s ∈ A with payoffs ( α, γ, β, δ ). Since nosuch profile exists Γ( G ) is only-transitive. g he α, α, β, β γ, δ, δ, γf δ, γ, γ, δ β, β, α, α ( a, c, , ) g he γ, δ, δ, γ α, α, β, βf β, β, α, α δ, γ, γ, δ ( a, d, , ) g he δ, γ, γ, δ β, β, α, αf α, α, β, β γ, δ, δ, γ ( b, c, , ) g he β, β, α, α δ, γ, γ, δf γ, δ, δ, γ α, α, β, β ( b, d, , ) G ′ = { (cid:0) (12) ◦ (34); (cid:0) a bd c (cid:1) , (cid:0) c da b (cid:1) , (cid:0) e fh g (cid:1) , (cid:0) g he f (cid:1)(cid:1) , (cid:0) (13) ◦ (24); (cid:0) a bf e (cid:1) , (cid:0) c dh g (cid:1) , (cid:0) e fa b (cid:1) , (cid:0) g hc d (cid:1)(cid:1) , (cid:0) (14) ◦ (23); (cid:0) a bh g (cid:1) , (cid:0) c df e (cid:1) , (cid:0) e fc d (cid:1) , (cid:0) g ha b (cid:1)(cid:1) } That Γ( G ′ ) is only-transitive non-standard symmetric follows by the same argument used for Γ( G ).15 .12 Example: n -transitively non-standard symmetric 4-player game. g he α, β, β, β β, α, β, βf β, β, β, α β, β, β, α ( a, c, , ) g he β, β, α, β β, α, β, βf β, β, α, β α, β, β, β ( a, d, , ) g he α, β, β, β β, β, α, βf β, α, β, β β, β, α, β ( b, c, , ) g he β, β, β, α β, β, β, αf β, α, β, β α, β, β, β ( b, d, , ) G = { (cid:0) (1234); (cid:0) a bc d (cid:1) , (cid:0) c de f (cid:1) , (cid:0) e fh g (cid:1) , (cid:0) g ha b (cid:1)(cid:1) , (cid:0) (12); (cid:0) a bc d (cid:1) , (cid:0) c da b (cid:1) , (cid:0) e fe f (cid:1) , (cid:0) g hh g (cid:1)(cid:1) } Γ( G ) is n -transitive since h G i is n -transitive, and non-standard symmetric since there does not exist anyprofile where all players receive the same payoff. Only-transitive non-standard symmetric 6-player game that has a subgroup h G i iso-morphic to −−→h G i with h G i N = { id Γ } . k li , , , , , , , , , , j , , , , ,
14 15 , , , , , a, c, e, g, , ) k li , , , , , , , , , , j , , , , ,
26 27 , , , , , a, c, e, h, , ) k li , , , , ,
10 29 , , , , , j , , , , ,
25 4 , , , , , a, c, f, g, , ) k li , , , , ,
16 8 , , , , , j , , , , ,
31 16 , , , , , a, c, f, h, , ) k li , , , , , , , , , ,
17j 30 , , , , ,
30 6 , , , , , a, d, e, g, , ) k li , , , , ,
15 32 , , , , , j , , , , , , , , , , a, d, e, h, , ) k li , , , , ,
22 12 , , , , , j , , , , ,
13 2 , , , , , a, d, f, g, , ) k li , , , , ,
27 24 , , , , , j , , , , ,
21 10 , , , , , a, d, f, h, , ) k li , , , , ,
12 25 , , , , , j , , , , ,
23 31 , , , , , b, c, e, g, , ) k li , , , , ,
29 14 , , , , , j , , , , ,
11 26 , , , , , b, c, e, h, , ) k li , , , , ,
24 10 , , , , , j , , , , ,
28 16 , , , , , b, c, f, g, , ) k li , , , , , , , , , , j , , , , ,
17 4 , , , , , b, c, f, h, , ) k li , , , , ,
18 28 , , , , , j , , , , , , , , , , b, d, e, g, , ) k li , , , , , , , , , , j , , , , , , , , , , b, d, e, h, , ) k li , , , , ,
32 24 , , , , , j , , , , ,
19 8 , , , , , b, d, f, g, , ) k li , , , , ,
20 12 , , , , , j , , , , , , , , , , b, d, f, h, , ) G = { (cid:0) (14) ◦ (25); (cid:0) a bh g (cid:1) , (cid:0) c di j (cid:1) , (cid:0) e ff e (cid:1) , (cid:0) g hb a (cid:1) , (cid:0) i jc d (cid:1) , (cid:0) k ll k (cid:1)(cid:1) , (cid:0) (135) ◦ (246); (cid:0) a be f (cid:1) , (cid:0) c dg h (cid:1) , (cid:0) e fi j (cid:1) , (cid:0) g hk l (cid:1) , (cid:0) i ja b (cid:1) , (cid:0) k lc d (cid:1)(cid:1) } Since there does not exist any profile where the payoffs are equal and h G i is transitive, Γ is non-standardsymmetric. Now the payoffs for ( a, c, e, g, i, k ) are (1 , , , , , s ∈ A with payoffs (2 , , , , , h G i has order 12, which is equal to the order of h (14) ◦ (25) , (135) ◦ (246) i , hence h G i ∼ = h (14) ◦ (25) , (135) ◦ (246) i and h G i N = { id Γ } . Part of the time spent typesetting this paper was supported by a Tasmania Graduate Research Schol-arship (186). The author would especially like to express his gratitude towards Jeremy Sumner for16roof-reading early drafts and Des FitzGerald who allowed me to explore symmetric games for the thesiscomponent of my honours year at the University of Tasmania and who has been useful for advice sincethen.
References
Felix Brandt, Felix Fischer, and Markus Holzer. Symmetries and the complexity of pure Nash equilibrium.
Journal of Computer and System Sciences , 75(3):163–177, 2009.Shih-Fen Cheng, Daniel M Reeves, Yevgeniy Vorobeychik, and Michael P Wellman. Notes on equilibriain symmetric games. In
Workshop on Game Theory and Decision Theory , 2004.Partha Dasgupta and Eric Maskin. The existence of equilibrium in discontinuous economic games, I:Theory.
The Review of Economic Studies , 53(1):18–25, 1986.Joaquim Gabarr´o, Alina Garcia, and Maria Serna.
On the Complexity of Game Isomorphism , volume4708 of
Lecture Notes in Computer Science . Springer Berlin / Heidelberg, 2007.David Goforth and David Robinson.
Topology of × Games . Routledge, 2005.The GAP Group. GAP – Groups, Algorithms, and Programming, Version 4.7.0, 2013. URL .John Harsanyi and Reinhard Selten.
A General Theory of Equilibrium Selection in Games . MIT Press,1988.Alexander Hulpke. Constructing transitive permutation groups.
Journal of Symbolic Computation , 39(1):1–30, 2005.John Nash. Non-cooperative games.
The Annals of Mathematics, Second Series , 54(2):286–295, 1951.Bezalel Peleg, Joachim Rosenm¨uller, and Peter Sudh¨olter.
The canonical extensive form of a game form:Part I. Symmetries . Springer, 1999.Noah Stein. Exchangeable Equilibria.
Doctoral Thesis, MIT , 2011.Peter Sudh¨olter, Joachim Rosenm¨uller, and Bezalel Peleg. The canonical extensive form of a game form:Part II. Representation.
Journal of Mathematical Economics , 33(3):299–338, 2000.John von Neumann and Oskar Morgenstern.