Game Theory

In this small essay, the problem of co-operation in business relationships and some solutions will be discussed using game theory. There are some different kinds of business relationships. However, the relationship between the firm and its competitors will be mainly investigated instead of covering all of them 1. The reason is that it is easy to adapt to other ones.

Let's start by considering a famous story to understand the problem of co-operation in business. The story's called Prisoner's Dilemma.

John and Peter have been arrested for robbing the bank and placed in separate isolation cells. The police do not have sufficient evidence to convict them unless one of them confesses. If neither confesses, they are convicted of a minor crime and are sentenced to 2 months. If both confess, they spend 12 months in jail. However, if one confesses and the other does not, the confessor is released immediately but the other is sentenced to 18 months in jail - 12 for the crime and 6 for obstructing justice.

Peter (deny) Peter (confess)

John (deny) 2, 2 18, 0

John (confess) 0, 18 12, 12 NE

(NE - Nash Equilibrium 2)

With self-interest, both have incentive to confess. For example, to John, if Peter denies, John will be free immediately, otherwise his time in jail will be reduced from 18 months to 12 months. So it is better for John to confess no matter what Peter does. We call confessing dominant strategy which gives you a better payoff irrespective of what your opponent does. The situation is the same with Peter. The result is that both confess and both will be in jail for 12 months. However, they could be better-off if both deny. But as I will show, it is difficult to achieve this better outcome. The essential reason is the contradiction between self-interest and common-interest. It is very risky for John (or Peter) to deny because his opponent has strong incentive to confess.

As you can see from Prisoner's Dilemma, co-operation is not always easy despite its benefit. There are two solutions to encourage co-operation: commitment and repeated games. We will look more detail about these and find out in which conditions and how they can be used effectively.

Let's look at another one - a real world example. The Organization of Petroleum Exporting Countries, OPEC, was founded in 1960. But it dropped entirely on January 18, 1993. We can explain this by game theory. To make it simple, we will limit the number of countries by 2 and call them A and B 3. Every year, each country has to decide whether it should produce high or low output. To keep the price high, each country agrees to produce low output but because of high price it also has incentive to produce high to earn more profit. We summarize this by the following payoff matrix:

Country B (high output) Country B (low output)

Country A (high output) 80, 80 NE 120, 25

Country A (low output) 25, 120 100, 100

(NE - Nash Equilibrium)

There is dominant strategy for both A and B (produce high output) but they can get better payoff by producing low output and get 100 each. Once again, it is hard to reach this better outcome (the same reason as Prisoner's Dilemma). In fact, there are the effects of both commitment and repeated games in OPEC but it was still not successful. We will analyze why this happened and how to improve the outcome.

Firstly, OPEC decided to resolve the price maintenance problem by instituting a set of quotas, one for each of the 13 member countries. For instance, Saudi Arabia, the world's largest oil exporter, got the largest quota and Ecuador, the world's smallest oil exporter, got the smallest quota, etc. This is considered a commitment among OPEC members. Unfortunately, however, each member continually exceeded its quota. One of the reasons is that the commitment is not credible. So, what makes a commitment credible?

* Contract: it must specify punishment if a commitment is not honoured.

* Cut off Communications: makes an action truly irreversible.

* Burning Bridges: makes the action that approves commitment unique.

* Leave Outcome Beyond Your Control: the extreme punishment.

* Moving in Steps: each party moves small steps to reduce the size of the commitment.

Secondly, it would seem that one way for OPEC to escape this fate would be to exploit the efficiency-enhancing properties of repeated game equilibria. There are three different cases of repeated games:

a) It is finitely repeated game and the one-shot game has only one Nash equilibrium.

b) It is finitely repeated game and the one-shot game has at least two Nash equilibria.

c) It is infinitely repeated game.

But the outcome can be improved only in case b and case c. Unfortunately, two facts combine to place OPEC into case a. One is that member countries are rapidly running out of oil. Both Venezuela and Indonesia expect to run out entirely by 2010. This makes the game finite in a hurry. Because even if the countries act as if they are going to last forever, their oil is not. Second, there is only one Nash Equilibrium (as depicted in the figure). The only one equilibrium is the reason why all members cheat on their quotas each and every period. We will examine more deeply three cases of repeated games to explain above arguments.

a) (This situation is also called Chainstore Paradox) As it is finitely repeated game so we can suppose that the number of games is 100. Using backward inductive, we can reason like that: at the round 100, each country has no incentive to cooperate so all of them choose to produce high output. At the round 99, the only reason countries would want to cooperate is the hope of cooperation in future periods. But as we have already known, they would not cooperate in the round 100 so they would not cooperate in the round 99, either. Similarly to rounds 98, 97, ...2, 1 and the result is that all rounds of the game happen exactly the same an one-shot game.