Minmax payoffs of a location game

We consider a two-player, sequential location game, with n stages. At each stage, players 1 and 2 choose locations from a feasible set in sequence. After all moves are made, consumers each purchase one unit of the good from the closest location. Since player 1 has a natural first-mover disadvantage here (player 2 can obtain a payoff of 12 just by replicating player 1’s moves), we examine her minmax payoff. When the number of stages is known to both players we show that (i) if the feasible locations form a finite set in R, player 1 must obtain at least 1 d+1 in the single-move game (ii) in the original Hotelling game (uniformly distributed consumers on the unit interval), player 1 obtains 1 2 even in the multiple stage game. However, player 1’s minmax payoff suffers if she does not know the number of moves, but player 2 does. In the Hotelling game, where the number of stages is either 1 or 2, player 1’s payoff falls to 5 12 . If she has no information at all about n, we provide a lower bound for her minmax payoff: it must at least equal half the payoff of the single-stage game. Computer Science Department, E-mail: [email protected], Tel: 412-268-3564 GSIA, E-mail: [email protected], Tel: 412-268-5744 GSIA, E-mail: [email protected], Tel: 412-268-3694 GSIA, E-mail: [email protected], Tel: 412-268-6895