VARIATIONS ON SIMPLE PAYOFF MATRICES

            Given any payoff matrix, the standard assumption is that the players choose their strategies simultaneously (in any event, that each player must choose a strategy in ignorance of the strategic choice of the other player — or, as described in the original handouts, by “secret ballot”) and without pre-play communication.

            However, it is enlightening to consider variations on this standard setup such as those outlined below. It is important to note that none of the following questions have unconditionally correct. That is, in each case the correct answer depends on the nature of the strategic situation, as specified by the payoff matrix (i.e., whether the game is Pure Coordination, Battle of Sexes, Battle of Bismark Sea, D-Day, Prisoner’s Dilemma, Chicken, etc.). For simplicity, suppose the payoff matrix is 2 × 2, i.e., there are two just two players, each with a choice between two strategies, so the game has just four possible outcomes (cells in the matrix). And suppose that the payoff matrix is common knowledge to each player, i.e., each player knows what the other player’s payoffs (interests/preferences/values are).

1.         Suppose that the players choose simultaneously but can engage in unrestricted pre-play communication before choosing their strategies. Does this affect their strategic choices? Does either player have an incentive to communicate his intentions truthfully? Would a message necessarily be believed by the other player?

2.         Continue to suppose that the players choose simultaneously but that limited pre-play communication is possible — specifically that one player can send a single one-way message to the other player (who cannot reply) before they choose their strategies. Would the privileged player send such a message? Would it be truthful? Would it be believed by the other player?

3.         Continue to suppose that the players choose simultaneously but that one of them may acquire strategic intelligence, i.e., may somehow “find out” the other player’s strategy in advance. Is such strategic intelligence always useful? Can it ever be harmful? What might the other player do if he discovers that his strategic plans have been “found out”? Might the other player want to have his plans “found out”?

4.         Continue to suppose that the players choose simultaneously but that one of them may (attempt to) engage in strategic deception, i.e., may allow the other player to apparently “find out” his strategy in advance but this information is misleading. Is it always advantageous to deceive the other player in this way? Can it ever be harmful? What might the other player do if he discovers that you are attempting to deceive him?

5.         Continue to suppose that a player can use the opportunity for pre-play communication to (somehow) convey a credible or irrevocable unconditional commitment to a strategy choice. Might such a player commit himself to a different strategy than he would otherwise choose? Will this advantage the player who makes the commitment? Might it advantage the other player also?

6.         Now suppose that the players make their strategic choices sequentially, rather than simultaneously? (The produces a game with perfect information.) Note that this entails two variants of a given (2 × 2) matrix: (i) player 1 makes the first move, and (ii) player 2 makes the first move. Do the fact that moves are made sequentially affect the choice that either player makes? Does it affect the outcome of the game? If so, does the advantage go to the first-mover or the second-mover? Might both players benefit, or be hurt, as a result of sequential moves?

7.         Continue to suppose that the players make sequential choices but that the second-mover can use the opportunity for pre-play communication to (somehow) convey (somehow) make (and communicate) an credible or irrevocable conditional commitment to a strategy choice, i.e., if you (the first-mover) choose your strategy X, I will choose my strategy Y. Might this player conditionally commit himself to a different strategy than what he would otherwise choose? Will this conditional commitment take the form of a threat or a promise? Will this advantage the player who makes the conditional commitment? Might it advantage the other player also?

8.         Suppose the players can use the opportunity for pre-play communication to negotiate and enter into a binding agreement as to what strategy each will chose and (perhaps) to reallocate their joint payoffs in some agreed upon manner, i.e., to make sidepayments? Does this affect their strategy choice and the outcome of the games?