Example
In order to see the basic idea, consider the following game.
S 0,0 1,3
B 3,1 0,0
B S
1
Out In
(2,2)
In this game, player 1 has option of staying out and getting the payoff of 2, rather
than playing the battle of sexes game with player 2. Now the battle of sexes has three
Nash equilibria: the pure strategy equilibria (B,B) and (S,S), and the mixed strategy
equilibrium ((3/4,1/4),(1/4,3/4)), where player 1 (resp., 2) plays strategy B (resp, S)
with probability 3/4. These equilibria lead to three subgame-perfect equilibria in the
larger game:
1. Player 1 plays In and then they play (B,B);
2. Player 1 plays out, but they would play (S,S) if player 1 played In, and
1
3. Player 1 plays out, but they would play the mixed-strategy equilibrium ((3/4,1/4),(1/4,3/4))
if player 1 played In.
Let us look at the last two equilibria closely. In these equilibria, after seeing that
player 1 has played In, player 2 believes that player 1 will play S with positive probability
(namely with probabilities 1 and 1/4 in equilibria in 2 and 3, respectively, above). In
other words, after seeing In, player 2 thinks that player 1 plays the strategy InS with
positive probability. But notice that this strategy is strictly dominated by staying out
(i.e., by the strategies OutB and OutS). Hence, after observing that player 1 has played
In, Player 2 comes to think that Player 1 may be irrational. Notice, however, that playing
In does not provide any strong evidence for irrationality of Player 1. Player 1 might have
played In with the intention of playing B afterwards, thinking that player 2 will also play
B. That is, after seeing In, Player 2 has revised his beliefs about the rationality of Player
1, while he could have revised his beliefs about Player 1’s intentions and beliefs. That
means that he did not believe in the rationality of Player 1 strongly enough. Had he
believed in the rationality of player 1 strongly, after seeing In, he would become certain
that player 1 will play B, and thus he would also play B. Therefore, if player 1 had
sufficient confidence in that player 2 strongly believes that player 1 is rational, then she
would anticipate that he would play B, and she would play In. Therefore, the last two
equilibria cannot be consistent with the assumptions (i) that players “strongly believe”
in players’ rationality and (ii) that they are certain that players “strongly believe” in
players’ rationality.
1
The argument in the previous paragraph is a froward induction argument, as it is
based an the idea that after seeing a move players must try to think about what the
other players are trying to do, and interpret these moves as parts of a rational strategy
if possible. In this way, forward induction introduces two important notions into the
analysis:
1. Context: In analyzing a game, one should not take the game in isolation, but
should rather determine the larger context in which the game is being played.
For example, the analysis of the battle of sexes may change dramatically, once we