Classical Game Theory and the Missile Crisis

Introduction

Game theory is a branch of mathematics concerned with decision-making in social interactions. It applies to situations (games) where there are two or more people (called players) each attempting to choose between two more more ways of acting (called strategies). The possible outcomes of a game depend on the choices made by all players, and can be ranked in order of preference by each player.

In some two-person, two-strategy games, there are combinations of strategies for the players that are in a certain sense "stable". This will be true when neither player, by departing from its strategy, can do better. Two such strategies are together known as a Nash equilibrium, named after John Nash, a mathematician who received the Nobel prize in economics in 1994 for his work on game theory. Nash equilibria do not necessarily lead to the best outcomes for one, or even both, players. Moreover, for the games that will be analyzed - in which players can only rank outcomes ("ordinal games") but not attach numerical values to them ("cardinal games") - they may not always exist. (While they always exist, as Nash showed, in cardinal games, Nash equilibria in such games may involve "mixed strategies," which will be described later.)

The Cuban missile crisis was precipitated by a Soviet attempt in October 1962 to install medium-range and intermediate-range nuclear-armed ballistic missiles in Cuba that were capable of hitting a large portion of the United States. The goal of the United States was immediate removal of the Soviet missiles, and U.S. policy makers seriously considered two strategies to achieve this end [see Figure 1 below]:

A naval blockade (B), or "quarantine" as it was euphemistically called, to prevent shipment of more missiles, possibly followed by stronger action to induce the Soviet Union to withdraw the missiles already installed.

A "surgical" air strike (A) to wipe out the missiles already installed, insofar as possible, perhaps followed by an invasion of the island.

The alternatives open to Soviet policy makers were:

Withdrawal (W) of their missiles.

Maintenance (M) of their missiles.

Soviet Union (S.U.)

Withdrawal (W)

Maintenance (M)

UnitedStates(U.S.)

Blockade(B)

Compromise(3,3)

Soviet victory,U.S. defeat (2,4)

Air strike(A)

U.S. victory,Soviet defeat(4,2)

Nuclear war(1,1)

Figure 1: Cuban missile crisis as Chicken

Key: (x, y) = (payoff to U.S., payoff to S.U.)4=best; 3=next best; 2=next worst; 1=worstNash equilibria underscored

These strategies can be thought of as alternative courses of action that the two sides, or "players" in the parlance of game theory, can choose. They lead to four possible outcomes, which the players are assumed to rank as follows: 4=best; 3=next best; 2=next worst; and l=worst. Thus, the higher the number, the greater the payoff; but the payoffs are only ordinal, that is, they indicate an ordering of outcomes from best to worst, not the degree to which a player prefers one outcome over another. The first number in the ordered pairs for each outcome is the payoff to the row player (United States), the second number the ...