The Model

Symmetry as a property of a mathematical object refers to its invariance under a certain type of transformation. Symmetries of games usually mean invariance of the payoffs under automorphisms of the set of action profiles induced by some group of permutations of the set of players. Since such an automorphism preserves the number of players that play a particular action, a characteristic feature of symmetries in games is the inability to distinguish between other players. The most general class of games with this property will be called anonymous. Four different classes of games are obtained by considering two additional characteristics: identical payoff functions for all players and the ability todistinguish oneself from the other players. The games obtained by adding the former property will be called symmetric, and presence of the latter will be indicated by the prefix “self”. For the obvious reason, we will henceforth talk about games where the set of actions is the same for all players, and write A= A₁ = A₂ = · · · = A_n and k = |A|, respectively, to denote this set and its cardinality.

Let Γ be such a game. For any permutationπ :N→N of the set of players, let π⁰ : A^N →A^Nbe the permutation of the set of action profiles such thatπ⁰((a¹, a², . . . , aⁿ)) = (a^π(1), a^π(2), . . . , a^π(n)). Then,Γ isanonymousifp_i(a_N) =p_i(π⁰(a_N))for alla_N∈A^N, i ∈ N and all π with π(i) = i. Similarly, Γ is symmetric if p_i(a_N) = p_j(π⁰(a_N)) for all a_N ∈ A^N, i, j ∈ N and all π with π(j) = i. Finally, Γ is self-anonymous if p_i(a_N) =p_i(π⁰(a_N))for alla_N∈A^N,i∈N, andself-symmetricifp_i(a_N) =p_j(π⁰(a_N)) for alla_N ∈A^N,i, j∈N. Since π⁰ is an automorphism of the set of action profiles that preserves the number of players who play a particular action, an intuitive way to describe anonymous games is in terms of equivalence classes of the automorphism group ofπ⁰, using a notion introduced by Parikh (1966) in the context of context-free languages. Given a set A of actions, the commutative image of an action profile a_N ∈ A^N is given by

#(a_N) = (#(a, a_N))_a∈A where #(a, a_N) =|{i∈N:a_i=a}|. In other words, #(a, a_N) denotes the number of players playing action a in action profile a_N, and #(a_N) is the vector of these numbers for all the different actions. This definition naturally extends to action profiles for subsets of the players.

Definition 5.1 (anonymity). Let Γ = (N,(A_i)_i∈N,(p_i)_i∈N) be a normal-form game, A a set of actions such thatA_i=Afor all i∈N. Γ is called

anonymous if p_i(a_N) = p_i(a_N⁰ ) for all i∈ N and alla_N, a_N⁰ ∈A^N with a_i = a_i⁰ and #(a_−i) =#(a_−i⁰ ),

symmetric if p_i(a_N) =p_j(a_N⁰ ) for all i, j∈N and alla_N, a_N⁰ ∈ A^N with a_i = a_j⁰ and #(a−i) =#(a_−j⁰ ),

self-anonymous if p_i(a_N) = p_i(a_N⁰ ) for all i ∈ N and all a_N, a_N⁰ ∈ A^N with

#(a_N) =#(a_N⁰ ), and

5.2 ·The Model 61 anon

symm s-symm s-anon

Figure 5.1: Inclusion relationships between anonymous, symmetric, self-anonymous, and self-symmetric games

self-symmetric if p_i(a_N) = p_j(a_N⁰ ) for all i, j ∈ N and all a_N, a_N⁰ ∈ A^N with

#(a_N) =#(a_N⁰ ).

When talking about anonymous games, we write p_i(a_i, x_−i) to denote the payoff of playeriunder any action profilea_Nwith #(a−i) =x_−i. For self-anonymous games,p_i(x) is used to denote the payoff of player iunder any profilea_N with #(a_N) =x. It is easily verified that the class of self-symmetric games equals the intersection of symmetric and self-anonymous games, which in turn are both strictly contained in the class of anonymous games. An illustration of these inclusions is shown in Figure 5.1. Figure 5.2 illustrates the different payoff structures forn=3and k=2.

In terms of the above characterization, a game is anonymous if the payoff p_i(a_N) of player i ∈ N in action profile a_N depends, besides his own action a_i, only on the number #(a, a_−i) of other players playing each of the actions a ∈ A, but not on who plays them. If two players exchange actions, all other players’ payoffs remain the same.

For two-player games, anonymity does not impose any restrictions, because action sets of equal size can simply be achieved by adding dummy actions. A game is symmetric if it is anonymous and if the payoff function is the same for all players. Hence, if two players exchange actions, their payoffs are also exchanged, while all other players’ payoffs remain the same. Many well-known games like the Prisoner’s Dilemma, Rock-Paper-Scissors, or Chicken are examples of symmetric (two-player) games. Simple congestion games (Ieong et al., 2005) are an example for the multi-player case. In a self-anonymous game the payoff of each player depends only on the number #(a, a_N) of players playing each of the actionsa∈A,including the player himself. If two players exchange actions, the payoffs of all players remain the same. Matching Pennies is a self-anonymous two-player game, and voting with identical weights can be seen as an example for the multi-player case.

Finally, in a self-symmetric game the payoff is always the same for all players and stays the same if two players exchange actions. Self-symmetric games thus are a special case of so-calledcommon payoff games, in which all players get the same payoff. Obviously such games always have a pure equilibrium, namely an action profile with maximum payoff.

Other games guaranteed to possess a pure equilibrium, and the complexity of finding an equilibrium in these games, have been investigated by Fabrikant et al. (2004).

62 5·Anonymous Games

Γ₁: (·,·,·) (a, g, c) (a, b,·) (·, e, f) (·, b, c) (d, e,·) (d, ·, f) (·,·,·)

Γ₂: (a, a, a) (b, c, b) (b, b, c) (e, d, d) (c, b, b) (d, d, e) (d, e, d) (f, f, f)

Γ₃: (·,·,·) (a, b, c) (a, b, c) (d, e, f) (a, b, c) (d, e, f) (d, e, f) (·,·,·)

Γ₄: (a, a, a) (b, b, b) (b, b, b) (c, c, c) (b, b, b) (c, c, c) (c, c, c) (d, d, d)

Figure 5.2: Relationships between the payoffs of anonymous (Γ₁), symmetric (Γ₂), self-anonymous (Γ3), and self-symmetric (Γ4) games for n = 3 and k = 2. Players 1, 2, and 3 simultaneously choose rows, columns, and tables, respectively, and obtain payoffs according to the vector in the resulting cell. Each lower case letter stands for a payoff value, dots denote arbitrary payoff values. As an example for the separation of the different classes, Γ₁ is not symmetric if a 6= c and not self-anonymous if b 6= g. Γ₂ is not self-anonymous if b6=c. Γ₃ is not self-symmetric ifa6=c.

There are ^n+k−1_k−1

distributions ofnplayers amongkactions. Since these are exactly the equivalence classes of the set of action profiles forn−1players under the commutative image, an anonymous game can be represented using at most n·k· ^n+k−2_k−1

numbers.

In the following, we call theexplicit representation of an anonymous game the one that simply lists the payoffs for each of the above equivalence classes, and note that the explicit representation requires space polynomial in n if and only if k is bounded by a constant.

On the other hand, its size becomes super-polynomial in neven for the slightest growth ofk. Nevertheless, space polynomial in nmay still suffice to encode certain subclasses of anonymous games with a larger number of actions if we use an implicit representation of the payoff functions like a Boolean circuit. It is easy to see that for games with a constant number of actions, any encoding of a game that has size at least linear in the number of players and satisfies the basic assumptions of rational and efficient play made throughout the thesis, is equivalent to its explicit representation under polynomial-time reductions.

Interestingly, the ability to distinguish oneself from the other players does not increase the complexity of the pure equilibrium and iterated dominance problems when players only have two actions.