Question

Consider the case when a two-player zero-sum simultaneous game is converted to a sequential game in which one player moves and then the other player moves, having observed the first player’s move. The second mover in such a game always does at least as well in any subgame-perfect equiibrium of the sequential game as in any Nash equilibrium of the simultaneous game. Is this true or false and explain why

Answer 1

Yes, the statement is true.

Zero sum game is the one in which, with every set of stratergies, there is always a gain of one person and equal loss for the other person such that the total gain/loss is zero for the game.

For such a game, if sequential game is played, such that one player moves first and having observed first player's move, second player plays rationally as he /she would play in any nash equillibrium of the simultaneous game, then he/she would play the stratergy that would maximise his /her payoff. In such a situation with sequential games, the second player would be atleast as well as he/she was in Nash equilibrium of simultaneous game. For example:

In this game, simultaneously, there is no pure stratergy nash equilibrium, but given the sequential game, If player A choses 0, Player will be choosing 0, if A chooses 1, B would choose 1, thus Player B will be at least better off in sequential game.