Give a simple example of a game with an infinite number of s

Give a simple example of a game with an infinite number of strategies in which a player has no undominated strategies

Solution

Let’s suppose the player has M strategies altogether. Suppose all M of these strategies are dominated: for each strategy s, there is another strategy s’ that dominates s. Start with any one of the player’s strategies, and call it Strategy 1. It’s dominated by some other strategy; call that Strategy 2. Strategy 2 is dominated by some strategy; call that one Strategy 3. And so on. After at most M steps, we’ll have run out of new strategies that could dominate; so the strategy that dominates the one from the previous step will have to be one of the strategies we already considered. In other words, we’ll have a “cycle” of domination. But then, because of the Proposition proved below, each strategy in the cycle would dominate itself. But no strategy can ever dominate itself, because that would require a payoff in some cell to be larger than itself. So we have established that indeed the player’s strategies cannot all be dominated: assuming they’re all dominated leads to a conclusion that can’t be true. The following proposition was an essential step in the above argument (or “proof”): Proposition: If Strategy A dominates Strategy B, and Strategy B dominates Strategy C, then Strategy A has to dominate Strategy C. Proof: Against every configuration (or “profile”) of strategies by the other players, A yields our player at least as large a payoff as B, and B yields at least as high a payoff as C (because A dominates B, and B dominates C). Moreover, there is some profile against which A yields a strictly higher payoff than B, and also a profile against which B yields a strictly higher payoff than C; against each of these two profiles, then, A must yield a strictly higher payoff than C