Given a two-player zero-sum finite sequential game with no chance vertices or hidden information, I would like to know if finding a single subgame perfect equilibrium is a sufficient proxy for the entire set of such equilibria, if only the payoffs and not the strategy are relevant.
This much is clear to me; suppose a two-player zero-sum finite sequential game with no chance vertices or hidden information consists of just a single decision vertex and a payoff vector for each edge following it, i.e.,
Then clearly a subgame perfect equilibrium occurs iff the left payoff is maximized, thus all subgame perfect equilibria have that same maximized left payoff. And because the game is two-player and zero-sum there are no additional degrees of freedom for the right payoff to vary. So the answer is yes for a single vertex.
It is also clear to me that because the game is sequential and has no hidden information, players have no strict incentive to mix, and if a player decides to mix anyhow, then intuitively that decision should be causally irrelevant to the game and the other player, because the realization of the mixing distribution will be known by the time the next decision occurs.
Is there some way to argue an inductive step to complement the base case of a single vertex? Is this even a viable approach given that game trees are hierarchical rather than linear? Or is there some other argument to go about this?
This much is clear to me; suppose a two-player zero-sum finite sequential game with no chance vertices or hidden information consists of just a single decision vertex and a payoff vector for each edge following it, i.e.,
Then clearly a subgame perfect equilibrium occurs iff the left payoff is maximized, thus all subgame perfect equilibria have that same maximized left payoff. And because the game is two-player and zero-sum there are no additional degrees of freedom for the right payoff to vary. So the answer is yes for a single vertex.
It is also clear to me that because the game is sequential and has no hidden information, players have no strict incentive to mix, and if a player decides to mix anyhow, then intuitively that decision should be causally irrelevant to the game and the other player, because the realization of the mixing distribution will be known by the time the next decision occurs.
Is there some way to argue an inductive step to complement the base case of a single vertex? Is this even a viable approach given that game trees are hierarchical rather than linear? Or is there some other argument to go about this?