HE+ Economics Nash Equilibrium

Transcription

1 HE+ Economics Nash Equilibrium Nash equilibrium Nash equilibrium is a fundamental concept in game theory, the study of interdependent decision making (i.e. making decisions where your decision affects others and their decision affects you). We will look at Nash equilibrium in a setting where the game is only played once and all players make their decisions at the same time. Nash equilibrium is defined as the situation where all players are playing their best decision possible, taking other players decisions as given. No-one can improve their outcome by deviating (changing their decision). The Prisoner s Dilemma Nash equilibrium is best seen through an example. Consider two prisoners in separate cells. They have both committed a crime and both have the decision of whether to confess their crime or deny it. If both deny the crime they will both be sentenced to 3 years in prison. If both confess to the crime, they will both get 2 years in prison a lenient sentence reflecting their honesty. If one confesses and the other denies, the confessor will get a very lenient sentence of 1 year, while the denier will get a severe sentence of 5 years. Both players want to minimise the time they spend in prison. We represent the game in a payoff matrix. We assign a number to each outcome. A higher number indicates a better outcome. The values of the numbers don t matter, as they are only used to order outcomes. The order does matter. Let s say that a 1-year sentence has a value of 3, a 2-year sentence a value of 2, a 3-year sentence a value of 1 and a 5-year sentence a value of 0. We write the payoffs as a vector (a,b), where a is Prisoner A s payoff and b Prisoner B s payoff. Our matrix is as follows: Confess (1,1) (3,0)

2 Now we need to find each prisoner s best responses to the other s decision. Let s imagine that Prisoner A knows that Prisoner B will confess. A is now choosing between a payoff of 1 if he confesses and 0 if he denies. His best response is to confess. We can represent the best response for A to a given decision by B with a *: Confess (1*,1) (3,0) Likewise, if B denies, A s best response is to deny. Completing the same exercise for B yields: Confess (1*,1*) (3,0) As the Nash equilibrium is where both players play their best response given the other player s decision, it is where both elements of the payoff vector have a * next to them. In the above game, there is one Nash equilibrium and it s where both confess. While (Deny, Deny) gives a better outcome for both prisoners, it is not a credible outcome. If one prisoner knows the other will deny, there is a clear incentive to confess. If the game were played many times, it may be possible to sustain a Nash equilibrium where both players deny each time they play the game if both adopt strategies that reward previous cooperation (denial). Games can have more than one Nash equilibrium. In this case, it is not always clear which equilibrium we would expect to see. You may expect the most reasonable one, but such a definition is subjective and it may not be possible to find suitable criteria to define reasonable in any case. Consider the game of two people walking directly towards each other on the pavement. Deviating from your path is annoying, but walking into each other is worse. If both people deviate, they will still walk into each other. We can represent the game as Person 1/Person 2 Deviate Walk straight Deviate (0,0) (1*,2*) Walk straight (2*,1*) (0,0)

3 The Nash equilibrium is where one person deviates and the other walks straight, but it s not clear who should deviate. In real life, one person may step to the side ostentatiously before the other does. This would make it clear what Nash equilibrium to expect in that context. Nash equilibrium in a duopoly A duopoly is a market with only two producers. Unlike in a perfectly competitive market, both producers now have market power. They will therefore consider the impact of their actions on their competitors and the impact of their competitor s actions on them. Such a market is best studied in a game theory context. Cournot duopoly Consider a market where two firms both have their own profit given by π i (q i, q j ) = [a b(q i + q j ) c]q i, where a c, i = 1,2 and j i. We consider firm 1 s problem (i.e. set i=1 and j=2). Firm 1 has not control over q 2, so it must treat it as fixed, i.e. as a constant. Firm 1 s problem is to maximise π 1 treating q 2 as a constant. From your Year 12 (or equivalent) maths course, you should know that you do this by finding the derivative dπ 1 /dq 1 and setting this equal to zero. (It s easier if you expand the brackets first.) We have π 1 = q 1 bq 1 2 bq 1 q 2, so, treating q 2 as a constant, dπ 1 dq 1 = 2bq 1 bq 2. Setting this derivative equal to 0 and solving for q 1 yields q 1 = 1 2b 2 q 2. This tells us the best (profit-maximising) q 1 in terms q 2 and is called firm 1 s best response or reaction function. Firm 2 has an identical problem, so q 2 = 1 2b 2 q 1.

4 Because Nash equilibrium requires both firms to be playing best responses to each other, we can substitute the expression for q 2 (firm 2 s best response function) into the expression for q 1 (firm 1 s best response function) and solve to find the Nash equilibrium level of firm 1 output (i.e. the Nash equilibrium q 1 ). We can do something similar to find the equilibrium q 2. Doing so yields q 1 = q 2 = 3b. The fact that both firms produce the same in equilibrium is unsurprising as both firms face the same profit-maximisation problem. Since both firms produce the same in equilibrium, the equilibrium is called a symmetric equilibrium. We can take a short cut to finding the q 1 and q 2 in symmetric equilibrium by using q 1 = q 2 in firm 1 s reaction function (i.e. we replace the q 2 with a q 1 ). The optimal q 1 is then the optimal q 2 by definition. This trick is especially useful with more than 2 firms and the above algebra becomes tedious. Substituting the optimal q 1 back into firm 1 s profit function, we find that π 1 = 2 9b Firm 2 s profit is the same. Under perfect competition, these profits would be zero. Here, > 0. both firms use their market power to make a positive economic profit. Betrand duopoly If, instead of choosing a quantity to produce and letting the market decide the price (a step skipped above, for simplicity), firms chose a price to sell at and let the market decide the quantity, there would be a different outcome. Consider a market in which only two firms operate. They both sell the same product. Clearly, consumers will buy the product from whichever firm sells it for the lower price. If both firms are charging the same price, we assume half of consumers buy from firm 1 and the other half from firm 2. Assume the product costs c to produce. 1 Therefore neither firm will charge less than c: it s better to sell nothing than to sell an amount greater than zero at a loss. 1 I.e. there are constant marginal costs of c and zero fixed costs. If you haven t come across these terms yet, don t worry. It s not important.

5 Now let s say firm 1 charges p 1 and firm 2 p 2, with p 1 > p 2 > c At the moment, firm 1 makes no profit, as everyone buys from firm 2. Firm 2 makes (p 2 -c), which is > 0, per unit it sells. The number of units sold is > 0 so total profit is > 0. Firm 1 should then charge one penny less than p 2 to make (p 2 -c-0.01), which is > 0, per unit it sells. That way, firm 1 makes a profit > 0. But now firm 2 doesn t make a profit. Now firm 2 should charge one penny less than what firm 1 is now charging, i.e. charge (p 2 -c-0.02). This process continues until both firms charge c This way both firms make a strictly positive profit and it is a unique Nash equilibrium. It is the only pair of prices p 1 and p 2 such that neither firm can improve its profits by charging a different price. If one firm chose to reduce prices further, that firm would make a zero profit, as it would if it raised its price from c The choice of 0.05 here is arbitrary, but you will see why something of this sort is necessary for the argument below.