1. What games do economists play? To see more of our products visit our website at www.anforme.co.uk Tom Allen, Head of Economics, Eton College

2. Playing games and Economics Any player who has played any form of game will know that there is interdependence between players. Each must consider the action and reaction of other players when making his own decision. Economists play their own games, with a view to maximising the utility of consumers, the profits of firms and the income of workers.

3. The prisoner’s dilemma 1 The Prisoner’s dilemma model is the starting point for most game theory. A crime has been committed. Two suspects, Butch and Sundance, are held captive in separate cells and cannot communicate with each other. Each prisoner is questioned and made aware of the potential prison term he could serve, dependent on whether he, or the other subject, confesses. G

4. A – both suspects confess to the crime and both get 6 years in prison. C There are 4 possible outcomes, each represented by a quadrant A to D in the figure below. B – Butch confesses and implicates Sundance, who does not confess. Butch gets just 1 year for ‘plea bargaining’ but Sundance gets 10 years. C C – a reversal of quadrant B. C D – Neither Butch nor Sundance confesses and each receive a low sentence for a lesser misdemeanor. C The prisoner’s dilemma 2

5. If both had denied the crime each would have received just 2 years. C The answer is A: both will confess and receive a sentence of 6 years. C What is the chosen outcome? C Both confess because they have no opportunity to discuss strategy with each other, and even if they had they would not have trusted each other to stick to their word. C The prisoner’s dilemma 3

6. Butch confesses, which is a dominant strategy for him. C It is his best choice regardless of Sundance’s choice. C Sundance will also confess leaving us with the (6,6) outcome. C This is a so-called ‘Nash equilibrium’ since neither player is willing to change his decision, regardless of whether the other player changes his mind. C The prisoner’s dilemma 4

7. Game theory in Economics 1 Assume a 2-firm model – a duopoly. C These 2 rival firms are considering their pricing strategies: should they raise or lower their price? C The pay off matrix in the next slide reveals profit figures. C Each firm is trying to maximise its profit. C We will assume that demand is price inelastic for the industry as a whole. C

8. Four outcomes are possible in the matrix below. C A – both price high and hence both gain high profit of £ 6k. C B – X prices high but is undercut by Y, thus X’s profits fall to £ 1k, but Y gains £ 10k. C C – a reversal of quadrant B. C D – both price low and both make low profits of just £ 2k each. C Game theory in Economics 2

9. The answer is that both firms will price low since, whichever strategy is chosen by the other player, each has a dominant strategy of pricing low. C This is even though both would benefit from a high price; the outcome is forced by a lack of trust. C Firms being aware of this have a temptation to collude and form an agreement to the detriment of consumers. C Thus most western governments punish collusion such as price-fixing. C Punishments centre around game theory as the authorities will give leniency to the firm which co-operates, especially if it implicates others. C Game theory in Economics 3 Game theory can be applied to a whole range of decisions by firms. C

10. Further economic applications 1 Consider the market failure of environmental pollution and the resultant global warming. C The atmosphere can be considered a common resource – degradation by one country will affect all others. C Environmental pollution can occur due to imperfect information on the part of polluters, the expense in reducing emissions and the desire of any single country to free ride on others. C This will lead to the tragedy of the commons where each individual will deplete the resource because he gains more utility from doing so than from refraining, whilst others continue to deplete the resource as well. C

11. The pay off matrix below shows why the rational action of any individual government is to pollute. C China and the USA are the two largest emitters of greenhouse gases. C The model assumes that the cost of either country reducing emissions is 5 negative units of utility and the benefit to both countries from either cutting emissions is 6 positive units, although it will cost them 5 negative units each to cut their emissions. C Further economic applications 2

12. There are four possible outcomes. C A -Both countries pollute and experience overall utility of 0 units each. C B - USA continues to pollute but gains an extra 3 units because China is cutting emissions. China’s utility falls from 0 to 5 negative units due to the cost of cutting pollution but rises by 3 due to the reduction of global emissions, leaving it with 2 negative units. C C- the reversal of quadrant B. C Further economic applications 3 D – here both countries cut emissions. This costs each country 5 units, but they then gain 3 + 3 = 6 due to reduced global emissions, leaving them with +1 each. C

13. The dominant strategy is for both to pollute, giving a Nash equilibrium at A. C The trade-off is even worse for small countries, as the relative cost to them of cutting emissions remains high, but the potential reduction in global emissions from their action is minimal. C Thus it appears that continued environmental pollution looks inevitable, at least until pollution gets so bad that the benefit to any one country from cutting emissions exceeds the costs of doing so. C Further economic applications 4 In the previous example it would require for China or USA the benefit of cutting pollution to rise from 3 units to a figure above 5 but by which time irreversible damage might have been done. C

14. Life is not so simple in reality as the examples given above. C The pay-offs might not result in a dominant strategy for both players leaving an unpredictable outcome. Also, in real life, many games are repeated. C Two firms might initially collude in Round 1, but then one firm may decide to undercut the other and scoop all the profits in Round 2. This might then lead to retaliation in Round 3. C Repeat games Empirical evidence shows that in repeated games like these, players are likely to co-operate with each other, rather than compete by undercutting as they begin to trust each other. C Perhaps this does offer hope for reducing environmental pollution after all. C