Consider the following two-player game: each player i has a value vi that the mechanism knows but the other player does not. each player simultaneously and independently tells the mechanism one of two actions, "keep" or "switch." if both players choose "switch," then each player receives a payoff equal to the value of the other player. otherwise, each player i receives vi as payoff. suppose the values are independently chosen at random from a finite set of values v according to some distribution, not necessarily uniform. prove that in any bayesian-nash equilibrium for this game, the highest value for which any player switches is the lowest value that could be drawn with nonzero probability from v.