Now define a strategy profilesNofΓ by letting, for eachj∈N,sj=si0, and assume for contradiction that sN is not an equilibrium. Then there exists a player j ∈Nand some strategy tj ∈ ∆(A) for this player such that pj(s−i, ti) > pi(sN). Further, by definition of pi,j, pi,i(s−i0 , t) > pi,i(sN0
i), contradicting the assumption thatsN0
i is an equilibrium of Γi.
This result applies in particular to the case where both the number of actions and the neighborhood size are bounded. Since the pure equilibrium problem in symmetric graphical games is NP-complete even in the case of two actions, we have identified a class of games where computing a mixed equilibrium is computationally easier than deciding the existence of a pure one, unless P=NP. A different class of games with the same property is implicit in Theorem 3.4 of Daskalakis and Papadimitriou (2005). It should be noted, on the other hand, that the existence of a symmetric equilibrium does not in general extend to games that are not anonymous or in which players have different payoff functions.
6.7 Discussion
In this chapter we have completely characterized the complexity of deciding the existence of a pure Nash equilibrium in games with bounded neighborhoods. This problem is NP-complete in games with neighborhoods of size two, two actions, and two-valued payoff functions. For neighborhoods of size one it is NL-complete in general, and L-complete if additionally the number of actions grows only very slowly. Some additional cases become tractable for games that further satisfy the most restrictive type of anonymity considered in Chapter 5 within each neighborhood.
For the other types of anonymity, two neighbors again suffice for NP-hardness. While the construction used in the proof of Theorem 6.17 can be generalized to arbitrary neigh-borhoods of even size, it is unclear what happens for odd-sized neighneigh-borhoods. The extreme case when the neighborhood of every player consists of all other players yields or-dinary anonymous games, in which the pure equilibrium problem is in P when the number of actions is bounded. It remains open at which neighborhood size the transition between membership in P and NP-hardness occurs. Another open problem concerns the complexity of the mixed equilibrium problem in anonymous graphical games. A promising direction for proving hardness would be to make the construction of Goldberg and Papadimitriou (2006) anonymous. Finally, as suggested in Section 6.5, it would be interesting to study the complexity of generalized satisfiability problems in the presence of matchings.
Chapter 7
Quasi-Strict Equilibria
Criticism directed at Nash equilibrium has been a recurring theme in previous chapters.
In the remaining two chapters of the thesis we consider two more solution concepts, each of them trying to address a particular shortcoming.
Consider again the single-winner game introduced in Chapter 2, in which Alice, Bob, and Charlie select a winner among themselves using a protocol in which each of them has to raise their hand or not. The game is repeated in Figure 7.1. As we have already noted in Chapter 4, the only pure equilibrium of this game, in which Bob raises his hand while Alice and Charlie do not, is particularly weak. Both Bob and Charlie could deviate from their respective strategies to any other strategy without decreasing their chances of winning. After all, they cannot do any worse than losing. A similar property in fact applies to all pure equilibria of ranking games: there exists at least one player, namely the one ranked last in the equilibrium outcome, who receives his minimum payoff regardless of his choice of action.
To alleviate the effects of phenomena like this, Harsanyi (1973) suggests to impose the additional restriction that every best response of a player be played with positive probability. A Nash equilibrium satisfying this requirement is called a quasi-strict equi-librium.1 Quasi-strict equilibrium is a refinement of Nash equilibrium in the sense that the set of quasi-strict equilibria of every game forms a subset of the set of Nash equilibria.
This may also be beneficial with respect to another weakness of Nash equilibrium, its potential multiplicity. It turns out that the second equilibrium of the game in Figure 7.1, in which Bob raises his hand while Alice and Charlie randomize uniformly over their re-spective actions, is quasi-strict. Interestingly, Charlie plays a weakly dominated action with positive probability in this equilibrium.
Quasi-strict equilibria always exist in two-player games (Norde, 1999), but may fail to do so in games with more than two players. The game in Figure 4.9 on Page 48 shows that in fact, quasi-strict equilibria can already fail to exist in the very restricted class
1Harsanyi originally referred to quasi-strict equilibrium as “quasi-strong”. However, this term has been dropped to distinguish the concept from Aumann’s (1959) strong equilibrium.
127
128 7·Quasi-Strict Equilibria c1
b1 b2 a1 3 1 a2 1 2
c2 b1 b2
1 2
2 1
Figure 7.1: Single-winner game involving Alice (player 1), Bob (player 2), and Charlie (player3), repeated from Figure 4.1. The dashed square marks the only pure equilibrium, dotted rectangles mark an equilibrium in which Alice and Charlie randomize uniformly over their respective actions. The latter is the unique quasi-strict equilibrium of the game.
of single-winner games.2 In this chapter, we study the existence and the computational properties of quasi-strict equilibrium in zero-sum games, general normal-form games, and certain classes of anonymous and symmetric games. Section 7.3 focuses on two-player games, and it shown that quasi-strict equilibria, unlike Nash equilibria, have a unique support. We further give linear programs that characterize the quasi-strict equilibria in non-symmetric and symmetric zero-sum games. In Section 7.4 we turn to games with more than two players. We first identify new classes of games where a quasi-strict equilibrium is guaranteed to exist, and can in fact be found efficiently. An interesting example of such a class are symmetric games where every player has two actions. We then show that deciding the existence of a quasi-strict equilibrium in games with more than two players is NP-hard in general. This is in contrast to the two-player case, where the decision problem is trivial due to the existence result by Norde (1999).