recorded our discussion on this topic in April of last year.) I recall such games represented in my earliest readings on game theory in the form of a decision-payoff matrix. In a two-player game in which each player makes a single decision from among a finite number of choices, without knowledge of the other player's decision, the decision-payoff matrix labels the rows with one player's options and the columns with the other player's options. The corresponding cell for a given combination of decisions yields the payoff to both players.
These are the purest simultaneous-move discrete-strategy games - each player faces a single decision based on known payoff results of all possible decisions but without knowing the decision of the other player beforehand. A payoff matrix cross-indexing each player's decision options to yield their payoffs can represent the game in an abstract, analyzable way. For example, suppose the two players are hunters, and each has the option to cooperate to pursue a stag, or to go off independently to hunt a hare. This Stag Hunt game is discussed nicely by Sam Hillier on his Consulting Philosopher blog, which inspired me to explore it a little more closely in the context of our game theory book study. In the matrix below, I represent the decision options of each player as 'S' for "Stag" or 'H' for "Hare." Let's assume that if both players hunt for the stag, each is rewarded with enough meat for three days, but if one hunts for the stag alone, he receives nothing. A player who hunts for a hare is rewarded with enough meat for himself alone for one day.
|Stag Hunt payoff matrix|
|Prisoners' Dilemma payoff matrix|
In the interest of comparison, let's normalize the Prisoners' Dilemma to payoffs comparable to the Stag Hunt. I'll map values as follows:
- -25 => 0
- -10 => 1
- -3 => 2
- -1 => 3
|Normalized comparison of Stag Hunt and Prisoners' Dilemma|
In a later post I hope to explore the significance of communication, or signaling, among players in a game. Clearly players are motivated to communicate and foster trust in the Stag Hunt, whereas the Prisoners' Dilemma fosters suspicion and is ripe for betrayal.
|Stag Hunt in a winner-take-all format|
In general, when a given strategy is better for a player regardless of what action the opponent takes, that strategy is said to be dominant. Any strategy that is objectively worse than some other strategy, regardless of the opponent's action, is a dominated strategy. In the case of the winner-take-all Stag Hunt where one player's expected reward for the stag hunt is less than the hare, the hare (for that player) becomes the dominant strategy. The opponent's best strategy under that dominant strategy - to give up the stag and hunt for a hare as well - defines the Nash equilibrium for the game - the outcome that neither player can improve for themselves unilaterally.
More complex simultaneous move games (those with more than two choices for each player) can be simplified by the elimination of dominated strategies from consideration. Best-response analysis consists of each player identifying their best response to each of the opponent's strategies. If any strategy combination between players is a best response for both, that combination is a Nash equilibrium (even if neither strategy is dominant).
Some time ago, I tried a game theory approach to a three-player game, which involved multiple decision matrices among the players. The same general principles apply - best-response analysis identifies strategy combinations that result in a Nash equilibrium, as I found in my analysis of the three-player case.
|Coffee Shop 'C' Preferred By Both|
I happened to notice that this payoff matrix looks like that of the Prisoners' Dilemma (or even the Stag Hunt) except that neither player is rewarded for defecting unilaterally. Both are purely motivated to coordinate their decisions.
|Sally likes 'C' better; Harry prefers 'D'|
"Chicken" is a different kind of coordination game in which players drive cars straight toward each other with the payoff going to the player who stays the course if the other swerves to avoid the collision. The penalty for both staying the course, however, is significant. Coordination games like these typically motivate communication, or signaling, to influence an opponent's decision. I expect to discuss signaling and screening in later posts.
Even in the simplest simultaneous-move games, the configuration of the payoff matrix can drive very different player behavior. It will be interesting to uncover game design ramifications resulting from studying this theory, as Sam Hillier has done.