1. VARIATIONS ON SIMPLE PAYOFF MATRICES Topic #6

2. The Payoff Matrix Given any payoff matrix, the standard assumption is – that the players choose their strategies simultaneously, or in any event, that each player chooses a strategy in ignorance of the strategic choice of the other player, or, as described in the original Playing Games handouts, by “secret ballot,” and – without pre-play communication. However, it is enlightening to consider variations on this standard setup, such as the following.

3. The Payoff Matrix (cont.) The answer to the question of whether a given type of variation – makes a difference and – makes what kind of difference itself varies greatly according to the nature of the payoff matrix and the nature of the game it represents. – That is, whether the game is Pure Coordination, Battle of Sexes, Battle of Bismarck Sea, D-Day, Prisoner’s Dilemma, Chicken, etc. For simplicity, we continue to suppose that 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). We suppose also that the payoff matrix is common knowledge to each player, i.e., – each player knows what the other player’s payoffs (interests/preferences/values/goals are).

4. Sequential Choice with Perfect Information Suppose that the players make their strategic choices sequentially and openly, – producing a game with perfect information. – Note that this entails two variants of a given (2 × 2) matrix: player 1 makes the first move, and player 2 makes the first move. Does 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 (vs. simultaneous) moves?

5. Full Pre-play Communication Suppose that the players 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?

6. Limited Pre-play Communication Suppose that the players can engage in only limited pre-play communication, – 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?

7. Strategic Intelligence Suppose that one player gains strategic intelligence, i.e., – somehow “finds out” in advance the strategy chosen by the other player in advance. Is such strategic intelligence always useful to the player who gains it? Is it ever harmful to the player who gains it? What might the other player do if he discovers that his strategic plan have been “found out”? Might the other player want to have his strategic plan “found out”?

8. Strategic Deception Suppose that one player 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 may be 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?

10. Credible Unconditional Commitment Suppose that a player can use the opportunity for pre-play communication to (somehow) convey (perhaps by means of some overt strategic move) 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?

11. Threats and Promises Suppose that the players make sequential choices but that the second-mover can use the opportunity for pre-play communication to (somehow) convey 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 the second mover 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?

12. Cooperative Games and Side Payments Suppose the players can use the opportunity for pre-play communication – to negotiate and enter into a binding (or enforceable) agreement as to what strategy each will chose and – perhaps to reallocate their joint payoffs in some agreed upon manner, i.e., to make side payments? Does this affect their strategy choices and the outcome of the games?

13. Repeated Play Suppose that the game is iterated — that is, – the same players will play this game repeatedly. and – know that they will do so. Does this affect their strategy choice and the outcome of each play of the game? Does matter whether the number of iterations is known to the players? Can the players acquire reputations with respect to how they play? Will these reputations help them in games with other players.

14. An Unproblematic Zero-Conflict Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

15. A Zero-Conflict Coordination Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

16. A Coordination Game with Conflict of Interest Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

17. A Strictly-Determined Zero-Sum Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

18. A Non-Strictly Determined Zero-Sum Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

19. A Prisoner’s Dilemma Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects

20. A Chicken Game Sequential Choice Communication Intelligence Deception Commitment Threats Promises Binding Agreements Side Payments Reputation Effects