Game Theory

Game theory is a bag of analytical tools designed to help us understand the phenomena that we observe when decision-makers interact. The basic assumptions that underlie the theory are that decision-makers pursue well-defined exogenous objectives (they are rational) and take into account their knowledge or expectations of other decision-makers’ behaviour (they reason strategically). (Osborne and Rubinstein 1994) In short, game theory deals with any problem in which each player’s strategy depends on what the other players do.

Situations involving interdependent decisions arise frequently, in all walks of life. A few examples in which game theory could come in handy include:
● Friends choosing where to go have dinner

● Parents trying to get children to behave

● Commuters deciding how to go to work

● Businesses competing in a market

● Diplomats negotiating a treaty

● Gamblers betting in a card game

All of these situations call for strategic thinking - making use of available information to devise the best plan to achieve one’s objectives. Game theory simply extends this concept to interdependent decisions, in which the options being evaluated are functions of the players’ choices. The appropriate techniques for analyzing interdependent decisions differ significantly from those for individual decisions. Even for strictly competitive games, the goal is simply to identify one’s optimal strategy.

Sequential Games

To analyze a sequential game, first construct a game tree mapping out all of the possibilities. Then follow the basic strategic rule: “look ahead and reason back”:

1. Look ahead to the very last decision, and assume that if it comes to that point, the deciding player will choose his/her optimal outcome (the highest payoff, or otherwise most desirable result).

2. Back up to the second-to-last decision, and assume the next player would choose his/her best outcome, treating the following decision as fixed (because we have already decided what that player will pick if it should come to that).

3. Continue reasoning back in this way until all decisions have been fixed.

The only time you even have to think is if another player makes a “mistake”. Then you must look ahead and reason back again, to see if your optimal strategy has changed.

Simultaneous Games

Turning to simultaneous games, it is immediately apparent that they must be handled differently, because there is not necessarily any last move. Consider a simple, but very famous example, called the Prisoners’ Dilemma: two suspected felons are caught by the police, and interrogated in separate rooms. They are each told the following:

● If you both confess, you will each go to jail for 5 years.

● If only one of you confesses, he gets only 1 year and the other gets 20 years.

● If neither of you confesses, you each get 1 years in jail.

We cannot look ahead and reason back, since neither decision is made first. We just have to consider all possible combinations.                                                                                                    

The game table (also called a payoff matrix) clearly indicates if that the other prisoner confesses, the first prisoner will either get 10 years if he confesses or 25 if he doesn’t. So if the other prisoner confesses, the first would also prefer to confess. If the other prisoner holds out, the first prisoner will get 1 year if he confesses or 3 if he doesn’t, so again he would prefer to confess. And the other prisoner’s reasoning would be identical. There are several notable features in this game. First of all, both players have dominant strategies. A dominant strategy has payoffs such that, regardless of the choices of other players, no other strategy would result in a higher payoff. This greatly simplifies decisions: if you have a dominant strategy, use it, because there is no way to do better. Thus, as we had already determined, both prisoners should confess. Second, both players also have dominated strategies, with payoffs no better than those of at least one other strategy, regardless of the choices of other players. This also simplifies decisions: dominated strategies should never be used, since there is at least one other strategy that will never be worse, and could be better (depending on the choices of other players). A final observation here is that if both prisoners use their optimal strategies (confess), they do not reach an optimal outcome. This is an important theme: maximizing individual welfare does not necessarily aggregate to optimal welfare for a group. Consequently, we see the value of communication. If the two prisoners could only communicate, they could cooperate and agree to hold out so they would both get lighter sentences. But without the possibility of communication, neither can risk it, so both end up worse off.

Game theory is exciting because although the principles are simple, the applications are far-reaching. Interdependent decisions are everywhere, potentially including almost any endeavour in which self-interested agents cooperate and/or compete. Probably the most interesting games involve communication, because so many layers of strategy are possible.