Question
Given the payoffs in the matrix below, two players play a variant of the “Battle of Sexes”(BoS) game, in which each player chooses in between F and C. At first, player 1 makes a choice in between playing the standard BoS game (described in the matrix below) or giving a gift to the other player, which reduces all the payoffs of player 1 (described in matrix) by 1 unit, but does not affect the payoffs of player 2. Finally, they play one of the simultaneous BoS games, “standard game” or “game after gift”.
Derive the reduced strategic form of the game. Are there any weakly dominated strategies in the reduced form game? What does iterated elimination of weakly dominated strategies yield in the reduced form?
Solution:
Firstly we The game as an extensive form game. And find SPNe of this game.
In The standard game, there are 2 pure strategy Nash equilibrium (F,F and C,C) and 1 mixed strategy Nash equilibrium ((3/4, 1/4), (1/4, 3/4)).
In the game after gift, there are 2 pure strategy Nash equilibrium (F,F and C,C) and 1 mixed strategy Nash equilibrium ((3/4, 1/4), (1/4, 3/4)).
And I construct the followings
And (standard game-FF, FF) is SPNE.
So far, this answer matches with the solution manual. But after this point, solution manual constructs the following matrix.
But I don’t understand how to write the payoffs and strategies of the matrix. Please explain how to construct this matrix. Thanks a lot.
Best Answer
Recall that a strategy specifies the actions of the player in each decision node, even if when following this strategy he cannot get there. So for Player 1 there are 3 decision nodes: the first (choosing S or G) and then in each game an action (F/C). In total, he should have 8 strategies. Clearly, if he chose S then what he would have done in G doesn't matter, so it is possible to only look at reduce strategies which disregard what happens in nodes the game visits with 0 probability when following this strategy. To conclude, he has 4 strategies of the form: first choose S/G, then choose F/C in the one-shot game that was chosen. So his strategies are SF,SC,GF,GC, where SF stands for: choose S, then choose F.
Player 2 has 2 decision nodes, so his strategies should simply say what he does in each game. There are 4 of them, FF, FC, CF, CC, where FC for example is F in the left game, C in the right game.
The payoffs are according to the extensive form game. For example, in the cell that corresponds to (SF,FF), we go to the S game, and both players play F. This is the same in the (SF,FC) cell, because the C doesn't matter as the game G is not played.
It is left to fill the matrix and remove dominated strategies.