1. 17. Game theory G 17 / 1 GENERAL ECONOMICS 6
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2. 17.0 OVERVIEW G 17 / 2 17.1 OLIGOPOLY page G 17 / 3
17.2 JOHN NASH page G 17 / 4
17.3 GAME THEORY page G 17 / 5
17.4 EXERCISES page G 17 / 13
17.5 VOCABULARY page G 17 / 15
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3. G 17 / 3
OLIGOPOLY
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4. G 17 / 4
JOHN NASH
John Forbes Nash Jr. (June 13, 1928 – May 23, 2015) was an American mathematician who made fundamental contributions to game theory.
Nash's work has provided insight into the factors that govern chance and decision-making inside complex systems found in everyday life.
His theories are widely used in economics. Serving as a Senior Research Mathematician at Princeton University during the latter part of his life,
he shared the 1994 Nobel Memorial Prize in Economic Sciences with
game theorists Reinhard Selten and John Harsanyi.
Sylvia Nasar's biography of Nash, A Beautiful Mind, was published in
1998. A film by the same name was released in 2001, directed by
Ron Howard with Russell Crowe playing Nash; it won
four Academy Awards, including Best Picture
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5. 17.3 GAME THEORY G 17 / 5 Comment:
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6. G 17 / 6
Prisoner’s Dilemma
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 is higher than if they both choose to deny any involvement in the crime.
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.
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7. G 17 / 7
Summary
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
Nash equilibrium
A strategy profile is a Nash equilibrium if no player can do better by unilaterally changing his or her strategy.
To see what this means, imagine that each player is told the strategies of the others. Suppose then that each player asks themselves: "Knowing the strategies of the other players, and treating the strategies of the other players as set in stone, can I benefit by changing my strategy?"
If any player could answer "Yes", then that set of strategies is not a Nash equilibrium. But if every player prefers not to switch (or is indifferent between switching and not) then the strategy profile is a Nash equilibrium. Thus, each strategy in a Nash equilibrium is a best response to all other strategies in that equilibrium.
What is the Nash equilibrium here?
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8. G 17 / 8 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 payoff matrix is shown below.
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9. G 17 / 9 PROFIT COMPANY B High production Low production
€ 5 mio, € 5 mio
€ 12 mio, € 4 mio
COMPANY A
€ 4 mio, € 12 mio
€ 10 mio, € 10 mio
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.
And what is the Nash equilibrium?
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10. G 17 / 10
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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11. G 17 / 11
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12. G 17 / 12
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13. G 17 / 13
17.4 EXERCISES
Analyze the following market situation using game theory:
An established firm and a newcomer to the market of fixed size have to choose the appearance for a product. Each firm can choose between two different appearances for the product; call them X and Y. The established producer prefers the newcomer's product to look different from its own (so that its customers will not be tempted to buy the newcomer's product) while the newcomer prefers that the products look alike.
We can model this situation using the following pay off matrix
NEW
COMPANY B X Y
EXISTING
(2,1)
(1,2)
COMPANY A
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14. G 17 / 14
Your solution:
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15. G 17 / 15
17.5 VOCABULARY EN NL
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16. G 17 / 16 EN NL
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