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Game Theory: Nash Equilibrium, Prisoner’s Dilemma

Game theory is concerned with predicting the outcome of games of strategy in which the participants (for example two or more businesses competing in a market) have incomplete information about the others’ intentions

  • Game theory analysis has direct relevance to the study of the conduct and behaviour of firms in oligopolistic markets – for example the decisions that firms must take over pricing and levels of production, and also how much money to invest in research and development spending.
  • Costly research projects represent a risk for any business – but if one firm invests in R&D, can a rival firm decide not to follow? They might lose the competitive edge in the market and suffer a long term decline in market share and profitability.
  • The dominant strategy for both firms is probably to go ahead with R&D spending. If they do not and the other firm does, then their profits fall and they lose market share. However, there are only a limited number of patents available to be won and if all of the leading firms in a market spend heavily on R&D, this may ultimately yield a lower total rate of return than if only one firm opts to proceed.

Nash Equilibrium

Nash Equilibrium is an important idea in game theory – it describes any situation where all of the participants in a game are pursuing their best possible strategy given the strategies of all of the other participants.

In a Nash Equilibrium, the outcome of a game that occurs is when player A takes the best possible action given the action of player B, and player B takes the best possible action given the action of player A

Two prisoners are held in a separate room and cannot communicate

 

They are both suspected of a crime

They can either confess or they can deny the crime

Payoffs shown in the matrix are years in prison from their chosen course of action

Prisoner A
Confess Deny
Prisoner B Confess (3 years, 3 years) (1 year, 10 years)
Deny (10 years, 1 year) (2 years, 2 years)
  • What is the best strategy for each prisoner?
  • Equilibrium happens when each player takes decisions which maximise the outcome for them given the actions of the other player in the game.
  • In our example of the Prisoners’ Dilemma, the dominant strategy for each player is to confess since this is a course of action likely to minimise the average number of years they might expect to remain in prison.
  • But if both prisoners choose to confess, their “pay-off” i.e. 3 years each in prison is higher than if they both choose to deny any involvement in the crime.
  • In following narrowly defined self-interest, both prisoners make themselves worse off
  • That said, even if both prisoners chose to deny the crime (and indeed could communicate to agree this course of action), then each prisoner has an incentive to cheat on any agreement and confess, thereby reducing their own spell in custody.
The equilibrium in the Prisoners’ Dilemma occurs when each player takes the best possible action for themselves given the action of the other player.

 

The dominant strategy is each prisoners’ unique best strategy regardless of the other players’ action

Best strategy? Confess?

A bad outcome! – Both prisoners could do better by both denying – but once collusion sets in, each prisoner has an incentive to cheat!

Prisoner A
Confess Deny
Prisoner B Confess (3 years, 3 years) (1 year, 10 years)
Deny (10 years, 1 year) (2 years, 2 years)

The Prisoner’s Dilemma

  • The classic example of game theory is the Prisoner’s Dilemma, a situation where two prisoners are being questioned over their guilt or innocence of a crime.
  • They have a simple choice, either to confess to the crime (thereby implicating their accomplice) and accept the consequences, or to deny all involvement and hope that their partner does likewise.

Confess or keep quiet? The Prisoner’s Dilemma is a classic example of basic game theory in action!

  • The “pay-off” in this game is measured in terms of years in prison arising from their choices and this is summarised in the table below.
  • No communication is permitted between the two suspects – in other words, each must make an independent decision, but clearly they will take into account the likely behaviour of the other when under-interrogation. This highlights the importance of uncertainty in an oligopoly.

Applying the Prisoner’s Dilemma to Business Decisions

  • Game theory examples revolve around the pay-offs that come from making different decisions.
  • In the prisoner’s dilemma the reward to defecting is greater than mutual cooperation which itself brings a higher reward than mutual defection which itself is better than the sucker’s pay-off.
  • Critically, the reward for two players cooperating with each other is higher than the average reward from defection and the sucker’s pay-off.

Consider this example of a simple pricing game:

The values in the table refer to the profits that flow from making a particular output decision. In this simple game, the firm can choose to produce a high or a low output. The profit payoff matrix is shown below.

  Firm B’s output
High output Low output
Firm A’s output High output £5m, £5m £12m, £4m
Low output £4m, £12m £10m, £10m
  • Display of payoffs: row first, column second e.g. if Firm A chooses a high output and Firm B opts for a low output, Firm A wins £12m and Firm B wins £4m.
  • In this game the reward to both firms choosing to limit supply and thereby keep the price relatively high is that they each earn £10m. But choosing to defect from this strategy and increase output can cause a rise in market supply, lower prices and lower profits – £5m each if both choose to do so.
  • A dominant strategy is one that is best irrespective of the other player’s choice. In this case the dominant strategy is competition between the firms.
  • The Prisoners’ Dilemma can help to explain the breakdown of price-fixing agreements between producers which can lead to the out-break of price wars among suppliers, the break-down of other joint ventures between producers and also the collapse of free-trade agreements between countries when one or more countries decides that protectionist strategies are in their own best interest.
  • The key point is that game theory provides an insight into the interdependent decision-making that lies at the heart of the interaction between businesses in a competitive market.

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