SHDSL – ITU-T G.991.2 [2]

4.4

Games With Non-Pure Equilibria

We noticed earlier that in the only pure equilibrium of the game in Figure 4.1, two of the three players have no reason whatsoever to stick to their equilibrium strategies. This in fact indicates an inherent weakness of pure Nash equilibrium as a solution concept for ranking games. In any outcome of a ranking game, some player is ranked last and receives his minimum payoff. This means that inany pure Nash equilibrium ,some player must be indifferent between all of his actions: no matter what he does, he will remain a loser. Obviously, the stability of a solution like that is highly questionable, if it is to be considered a viable solution at all. This is particularly true for single-winner games, where in a pure equilibrium all players but the winner are indifferent over which action to play.

On the other hand, it is very well possible that all the actions in the support of amixed equilibrium yield a strictly higher expected payoff than any action not in the support, mitigating the phenomenon mentioned above. Equilibria satisfying this condition are called quasi-strict, and will be treated in more detail in Chapter 7.

For now we concentrate on non-pure equilibria, i.e., equilibria in which at least one player randomizes. We conjecture that every single-winner game possesses at least one such equilibrium, and prove this claim for three subclasses.

Theorem 4.2. The following classes of ranking games always possess at least one non-pure equilibrium:

(i) two-player ranking games,

(ii) three-player single-winner games where each player has two actions, and (iii) n-player single-winner games where the security level of at least two players is

positive.

Proof. Statement (i) follows directly from the fact that every two-player game has a quasi- strict equilibrium (Norde, 1999), and the above observation that quasi-strict equilibria of ranking games are never pure. Here we give a simple alternative proof. Assume for contradiction that there is a two-player ranking game that only possesses pure equilibria and consider, without loss of generality, a pure equilibrium s∗_N in which player 1 wins. Since player 2 must be incapable of increasing his payoff by deviating from s∗_N, player 1

has to win no matter which action the second player chooses. As a consequence, the strategies in s∗_N remain in equilibrium even if player 2’s strategy is replaced with an arbitrary randomization among his actions.

As for (ii), consider a three-player single winner game with actionsA₁={a1, a2},A₂ = {b1, b2}, andA₃ ={c1, c2}. Assume for contradiction that there are only pure equilibria in the game and consider, without loss of generality, a pure equilibrium s∗_N = (a1, b1, c1)in which player1 wins. In the following, we say that a pure equilibrium issemi-strict if at least one player strictly prefers his equilibrium action over all his other actions, given that

36 4 ·Ranking Games

the other players play their equilibrium actions. In single-winner games, this player has to be the winner in the pure equilibrium. We first show that ifs∗_Nis semi-strict, i.e., if player1

does not win in action profile(a2, b1, c1), then there must exist a non-pure equilibrium. For this, consider the strategy profile s1_N = (a1, s₂1, c1), where s1₂ is the uniform mixture of player2’s actionsb1 andb2, along with the strategy profiles2_N = (a1, b1, s2₃), wheres2₃

is the uniform mixture of actions c1 and c2 of player 3. Since player 1 does not win in (a2, b1, c1), he has no incentive to deviate from either s1_N or s2_N, even if he wins in

(a2, b2, c1) and (a2, b1, c2). Consequently, player 3 must win in (a1, b2, c2) in order fors1_N not to be an equilibrium. Analogously, for s2_N not to be an equilibrium, player 2

has to win in the same action profile (a1, b2, c2), contradicting the assumption that the game is a single-winner game. The existence of a semi-strict pure equilibrium thus implies that of a non-pure equilibrium. Now assume thats∗_N isnot semi-strict. When any of the action profiles inB={(a2, b1, c1),(a1, b2, c1),(a1, b1, c2)}is a pure equilibrium, this also yields a non-pure equilibrium because two pure equilibria that only differ by the action of a single player can be combined into infinitely many mixed equilibria. For B not to possess any pure equilibria, there must be (exactly) one player for every profile inBwho deviates to a profile in C = {(a2, b2, c1),(a2, b1, c2),(a1, b2, c2)}, because the game is a single-winner game and because s∗_N is not semi-strict. Moreover, either player 1 or player 2 wins in(a2, b2, c1), player2 or player3 in(a1, b2, c2), and player1 or player3

in(a2, b1, c2). This implies two facts. First, the action profile s3_N= (a2, b2, c2)is a pure equilibrium because no player will deviate froms3_Nto any profile inC. Second, the player who wins in s3_N strictly prefers the equilibrium outcome over the corresponding action profile inC, implying thats3_N is semi-strict. The above observation that every semi-strict equilibrium also yields a non-pure equilibrium completes the proof.

As for (iii), recall that the payoff a player obtains in equilibrium must be at least his security level. A positive security level for playeri thus rules out all equilibria in which player ireceives payoff zero, in particular all pure equilibria in which he does not win. If there are two players with positive security levels, both of them have to win with positive probability in any equilibrium of the game. In single-winner games, this can only be the case in a non-pure equilibrium.

We conjecture that this existence result in fact applies to the entire class of single- winner games. To see this it doesnotextend to general ranking games, consider a4-player game in which the first three players have two actions. Payoffs for the case when player4

plays his first actiond1 are shown in Figure 4.4. We can in fact restrict our attention to this case by assigning payoffs(1, 0, 0, 0)to any action profile where player4plays one of his potential other actions, thereby making them strictly dominated. The resulting game is then similar to a3-player game by van Damme (1983) that does not have anyquasi-strict equilibria. It is furthermore straightforward to show that the game is a ranking game by virtue of rank payoff vectorsr1 =r3= (1, 0, 0, 0),r2 = (1,1₂, 0, 0), andr4 = (1, 1, 1, 0), and that the pure equilibria(a1, a2, a3) and (b1, b2, b3) are the only equilibria of this game.

4.5 ·Solving Ranking Games 37