Tit for Tat

Tit for Tat is a highly-effective strategy in game theory for the iterated prisoner's dilemma. It was first introduced by Anatol Rapoport in Robert Axelrod's 1984 tournament. Based on the English saying meaning "equivalent retaliation" ("tip for tap"), an agent using this strategy will initially cooperate, then respond in kind to a previous opponent's action. If the opponent previously was cooperative, the agent is cooperative. If not, the agent is not.

Contents

1 Overview 2 Example of Play 3 Implications 4 External link 5 References

Overview

This strategy is dependent on four conditions that has allowed it to become the most prevalent strategy for the Prisoner's Dilemma:

1. Unless provoked, the agent will always cooperate
2. If provoked, the agent will retaliate
3. The agent is quick to forgive
4. The agent must have a 2/3 chance of competing against the opponent more than once.

In the last condition, the number 2/3 is arbitrary and depends on the payoff matrix of the Prisoner's Dilemma. The important thing is that the competition continues long enough for repeated punishment and forgiveness to generate a long-term payoff higher than the possible initial loss from cooperating initially.

A fifth condition applies to make the competition meaningful: if an agent knows that the next play will be the last, it should naturally defect for a higher score. Similarly if it knows that the next two plays will be the last, it should defect twice, and so on. Therefore the number of competitions must not be known in advance to the agents.

For several decades Tit for Tat was the most effective strategy for playing the game, winning in annual automated tournaments against (generally far more complex) strategies created by teams of computer scientists, economists, and psychologists. Game theorists informally believed the strategy to be optimal (although no proof was presented).

Example of Play

Assume there are four agents: two are Tit for Tat players ("variables") and two are simply trying to maximize their own winnings ("controls"). Assume that each player faces the other three in a match lasting six games. If one player gives evidence against a player who does not, the former gains five points and the latter nets zero. If both refrain from giving evidence, both gain three points. If both give evidence against each other, both gain one point.

When a variable faces off against a control, the former refrains from giving evidence in the first game while the control does the opposite, gaining the control five points. In the remaining five games, both players give evidence against each other, netting one point each game. The final score is control, ten; variable, five.

When the variables face off against each other, each refrains from giving evidence in all six games. Six times three is eighteen points for each variable.

When the controls face off, each gives evidence against the other in all six games. Six times one is six points for both controls.

The final score for each variable is five plus five plus eighteen, or twenty-eight points. The final score for each control is ten plus ten plus six, or twenty-six points. Despite the fact that the variables never won a match and the controls never lost a match, the variables still came out ahead, because the final score is not determined by the winner of matches, but the scorer of points. Simply put, the variables gained more points tying with each other than they lost to the controls.

(This example was taken from Piers Anthony's novel, Golem in the Gears.)

Implications

The success of the strategy, which is largely cooperative, took many by surprise. In successive competitions various teams produced complex strategies which attempted to "cheat" in a variety of cunning ways, but Tit for Tat eventually prevailed in every competition.

Some theorists believe this result may give insight into how groups of animals (and particularly human societies) have come to live in largely (or entirely) cooperative societies, rather than the individualistic "red in tooth and claw" way that might be expected from individual engaged in a Darwinian struggle. This, and particularly its application to human society and politics, is the subject of Robert Axelrod's book The Evolution of Cooperation.

External link

• Wired magazine story about Tit for Tat being defeated (http://www.wired.com/news/culture/0,1284,65317,00.html)

References

• The Evolution of Cooperation, Robert Axelrod, Basic Books, ISBN 0465021212
• The Origins Of Virtue, Matt Ridley, Penguin Books Ltd, ISBN 0140244042de:Tit-for-tat

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