INTRO GAME THEORY
INTRO GAME THEORY ECON 171
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Handout on Nash Equilibrium Econ 171 Fall 2008 Each of the three games in Figure 1 isis not cross out one dominance solvable H T Battle of the Sexes Matching Pennies Figure 1 Two classic normal form games plus one De nition 1 A strategy pro le 8 E S is a Nash equilibrium and only si 6 BRisl for each player i That is uisi s4 2 uissi for each s E 51 and each player i Find the Nash equilibria of all the games in Figures 13 Prisoners7 Dilemma again Chicken Coordination Figure 2 Three more of the classic normalform games Pareto Coordination Pigs Figure 3 The last two classic games De nition 2 Consider a strategy pro le a 01 021 1 107 where 01 E ASZ for each player i Pro le U is a mixedstrategy Nash equilibrium if and only uiai 071 2 uio ai for each s E 51 and each player i That is al is a best response to 01 for every player i Is the following statement always true sometimes true or never true If given her beliefs a nonpure mixed strategy is optimal for a player then it is uniquely optimaliit yields a strictly higher expected utility than all other pure or mixed strategies77 This statement is 7 W true For given beliefs a mixed strategy is W7 for a player 7 W7 each action she plays with positive probability is optimal The implies that a mixed strategy is W7 uniquely optimali Now nd all the Nash equilibria of all the games in Figures 13 Two more to practice on the back Econ 171 Fall 2008 Lecture 16 November 20 Why SPNE is inadequate Recall how we originally motivated SPNE with this entry game 71771 21 Figure 1 The entry game F A In Out Figure 2 The entry game in normal form Figure 3 A variant of the entry game De nitionl A belief system in an extensive game is a function that assigns to each information set a prob ability distribution over the histories in that informa tion set De nition 2 An assessment in an extensive game is a pair consisting of a strategy pro le and a belief system 1 Econ 171 Fall 2008 Notes in Place of Lecture 5L October 9 Reminder these notes take the place ofleeture on Thursday October 9 The heat class will meet as scheduled on Tuesday October 14 Overview These notes explore some applications of Nash equilibrium and are intended to supplement Chapter 10 of Watson The rst section presents an extension of the Cournot and Bertrand duopoly models to more general oligopolistic settings It largely consists of a guided tour through Watson problems 101 and 102 The second section looks at strategic considerations in Political Economy It looks at the way politicians choose their platforms through the lens of a simple location model and analyzes voter turnout Finally7 the last section looks at strategic social situations7 such as public good provision and reporting crime Oligopoly One ofthe primary concerns of microeconomic theory is how market structure affects market outcomes and consumer and producer welfare The standard market models that we use to study the two extremes of perfect competition and monopoly provide help us understand how prices and quantities depend upon market structure and the conditions under which goods will be provided ef ciently by the market In reality7 the structure of many industries lies somewhere in between in these two extremes non monopolistic rms competing with each other7 but not so many that one rm can disregard the effect of its own choices on others7 or the impact of other rms7 behavior on its own outcomes Standard market models do not capture this interdependence7 so we turn to the tools of game theory to study oligopoly In previous microeconomic theory courses7 you may have studied simple Cournot and Bertrand duopoly models similar to those presented in Chapter 10 In this section we will make the leap from duopoly to more general oligopoly7 extending the Courtnot and Bertrand models to allow an arbitrary number of rms What we accomplish will be very close to solving Watson Problems 101 and 102 Cournot Suppose there are 71 rms that simultaneously and independently choose quantities7 with 11 2 0 being rm is quantity The price p is determined by inverse market demand7 which is given by p a 7 bQ7 where Q ELI qi and each rm faces a constant marginal cost 0 Assume that a gt e gt 0 and b gt 0 and that each rm seeks to maximize pro ts 1These notes rely heavily on Markus Moebius7 course materials and on Martin Osborne7s Introduction to Game Theary7 Oxford7 2004i First let us formalize this game by writing the strategy spaces and payoff functions For each 239 S 0 00 The payoff functions are written as pQq 7eql or Uiq Q a i 174139 bQ7iQi 0 Where Q7i Q Qi To analyze the model we solve for Nash equilibrium behavior We begin by deriving rm is best response function Firm 239 solves qagxla bQ7i bQilQi 0 which yields the rst order condition a 7 bQ 7 e 2bqi Solving for 11 reveals rm is best response function to be qlQ E 7 as represented in the graph below 212 1 ac2b acb Figure 1 Player is best response function Now we solve for the equilibrium quantity We start by noting the problems symmetry all players will produce the same output 1 in equilibrium so we denote total equilibrium output Qquot nqquot Thus Q Li n 7 1q Plugging this into the best response function yields 1 M Wlm 211 for a total industry output of Qquot Using we calculate the price to be pquot 0 then individual pro ts to be ul n1 z pq 7 cqquot and industry pro ts to be Uquot How does this result compare to the monopoly and perfect competition outcomes First we look at monopoly A monopolist is the only producer so Q 1 The monopolists maximizes pro ts by setting marginal revenue equal to marginal cost We derive marginal revenue by differentiating the expression RQ pQQ a 7 bQQ which yields MRQ a 7 2bQ Setting this equal to MC e and solving gives us Q96 as the optimal quantity and pquot as the optimal price The monopoly pro ts are U For perfect competition we use the fact that competition drives industry wide output high enough so that the price equals marginal cost so PQ a 7 bQ e or Q Each rm thus produces 11 and industry and rm pro ts are of course zero because price equals marginal cost These results give us a very clear picture of how market structure affects output prices and pro ts As an oligopolistic industry grows more competitive n grows each rm7s output decreases but the overall output grows As 71 a 00 each rm produces a vanishingly small amount and the total output converges to that of a perfectly competitive industry Con versely as n shrinks to 1 the output both rm and industry converges to the monopoly output The industry produces less but each rm having a larger share of the market produces more We see similar pattern when we look at price and pro ts The monopolists price is high est because the total quantity is lowest The monopolist enjoys all of the sizeable industry pro ts As more competitors appear the growth of total output pushes down the price In dustry pro ts still grow due to increased output but shrinking market share and decreasing price serve to reduce rm level pro ts As the industry approaches perfect competition the price converges to marginal goes and pro ts shrink to zero Now we see that monopoly and perfect competition are not two isolated and distinct phenomena to be studied separately but rather two extremes on a spectrum of market structures Bertrand The Bertrand model considers rms that compete on price not quantity Each chooses p 2 0 and the lowest price offered on market will be denoted p minp1 pn Consumers only purchase from the rm or rms charging p with demand given by Q a 7 p If multiple rms offer the lowest price these rms equally split the quantity demanded TAssume that rms must supply what is demanded of them marginal cost is c and a gt c gt 0 First we formalize the game by specifying the strategy spaces and payoff functions For each 239 S 0oo and Ha p39p39 7 C ifp p m l l l 7 MW 0 imp 7 where m denotes the number of players k E 1 2 n such that pk E For any n 2 2 p c for all 239 is a Nash equilibrium Any price above marginal cost cannot be the lowest price in equilibrium because other players would have an incentive to bid the price lower For n gt 2 there are other Nash equilibria in which one or more players selects a price greater than c but at least two players select 0 Take a moment to think about why such an equilibrium would work and why it does not work for n 2 One nal thing to note is that the notion of best response is not well de ned in this model Let 371 be the minimum 19 selected by any player j 31 239 lf 0 lt 8 player is best response is to select p lt 871 but as close as possible to 871 However there is no such number Political Economy Choosing a Platform Now we turn to political economy We begin by applying a location model similar to that presented in Chapter 8 of Watson to the question of politicians choosing their platform We model the population of voters as being uniformly distributed along the ideological spectrum from left x 0 to right x 1 One of ce is up for grabs and each candidate simultane ously chooses a campaign platform ie a point on the line between z 0 and z 1 The voters observe the candidates7 positions7 and then each voter votes for the candidate whose platform is closest to the voters position in the spectrum Suppose7 for example7 that there are two candidates and that they choose platforms 1 3 and x2 6 Then all voters to the left of z 45 would vote for candidate 17 all those to the right vote for candidate 27 and candidate 2 wins the election with 55 of the vote As a tie breaking rule7 assume that any candidates who choose the same platform equally split the votes cast for that platform7 and that ties among the leading vote getters are resolved by coin ips Also7 assume that the candidates care only about being elected7they do not really care about their platforms at all What does a pure strategy Nash equilibrium look like in this situation It depends upon the number of candidates Lets start simple with n 2 Both candidates locate at the center of the spectrum 1 2 05 Neither has an incentive to deviate The payoff from this strategy is ul uz 05 because each candidate captures exactly half the vote7 and ties are resolved by coin ips Without loss of generality assume that candidate 1 deviates from this play and moves a little bit to the left to 1 05 7 6 She now captures all the voters left of her and half the voters in the interval 05 7 6 05 She therefore gets a share 05 7 E g of the vote This is less than 50 and she will lose the election for sure The same applies if she moves to the right Therefore7 campaigning from the center is a Nash equilibrium If there are three candidates7 campaigning from the center is no longer a Nash equilibrium because by moving slightly to the left or right a candidate can capture almost half the votes and will win since her rivals get about 25 each One NE is if two players 12 locate a small distance 6 gt 0 to the left right and one player 3 locates E to the right left Player 37 the lonely one7 will surely win the contest and therefore has no incentive to deviate You should check that the two opponents 17 2 cannot win by deviating If one of them say 1 moves to a location z lt 127 the lone player will still win because he captures more than half of the vote If the opponent moves to z 12 then the remaining opponent 2 or the lone player 3 will win7 but certainly not the defector lf player 1 moves to a position gt 12 then player 2 will win for sure So no player has an incentive to deviate Voter Turnout Individuals may prefer one political candidate over another7 but the act of voting takes time and energy and only rarely does one7s vote actually affect which candidate is elected How do people decide whether or not to cast their vote How do we predict voter turnout will depend upon the size and makeup of the voting population Let us construct a simple game theoretic model to pursue these questions Suppose that two candidates A and B compete in an election Of the n citizens k support candidate A and m n 7 k support candidate B Each citizen decides whether to vote at a cost for the candidate she supports or to abstain A citizen who abstains receives the payoff of 2 if the candidate she supports wins 1 if this candidate ties for rst place and 0 if this candidate loses A citizen who votes receives the same payoff in each of these cases minus the cost 0 of voting where 0 lt c lt 1 Take a moment to think about this game when k m 1 Write out the matrix repre sentation What familiar game is this the same as2 What is the Nash equilibrium Both players vote which means the candidates tie If either deviates by not voting the c that she saves is outweighed by the 1 that she loses when she causes her candidate to lose Now let us consider the Nash equilibrium in the more general case that k m Do the candidates tie or does one candidate win Suppose that one candidate wins by a single vote In that case any non voting supporter of the losing candidate could pro tably deviate by voting and bringing her candidate into a tie On the other hand if one candidate wins by more than one vote any voting supporter of the winning candidate could pro tably deviate by not voting which saves c but does not affect her candidate7s victory By elimination we know that the candidates must tie but does everyone vote no one or do only some people vote If not everyone votes then any non voter can pro tably deviate by voting because that vote will bring victory out of a tie If everyone votes no one wants to switch because doing so would cause her candidate to lose Thus the unique Nash equilib rium is for everyone to vote Of course not everyone votes in real life In order to explain this perhaps we should consider a messier more realistic world One possibility is to let one candidate be favored by more people so k lt m without loss of generality In this case we can try to use the same kind of arguments that we used above Take a moment to do this Think of various possible scenarios a tie a victory for B no voting everyone voting all of A supporter7s voting and k 1 of Bs supporter7s voting etc In each of these would anyone have an incentive to deviate Would a non voter want to switch to voting in order to change the outcome or would voter want to not vote because it will not affect the outcome Eventually you should be able to see that there is no Nash equilibrium in pure strategies What can we say about voting turnout then Fortunately we can consider mixed strategies as well Consider the following mixed strategy pro le Of the supporters of candidate B k vote with certainty and m 7 k abstain while each of the k supporters of candidate A votes with probability p Let us verify that this is an equilibrium 2Prisoners7 Dilemmai Which strategy corresponds to C and which corresponds to D Does this seem strange to you Try to make some sense of it We start with the basic fact that in order for A supporters to be willing to mix they must be indifferent between voting and not voting How do we calculate the expected utility of each of these We consider each outcome and how likely it is to be realized It is impossible for A to win outright because exactly k supporters of B vote with certainty Similarly if an A supporter does not vote she guarantees that A will lose and she gets a payoff of 0 If she does vote she pays the cost 0 but with some probability she also seals a tie for A instead of a loss How likely is this to happen It requires all other A supporters to choose to vote as well which occurs with probability pk l The indifference equation is UAvote UAno vote 701pk71017pk71 0 19 will At this probability do any B supporters have an incentive to deviate First consider any of the k voters Voting means incurring cost 0 and winning in every case except when all ofthe A supporters randomly choose to vote causing a tie Thus UB votelvoter 7c21 710k pk On the other hand not voting leaves three possibilities With probability pk k supporters of A will vote and B will lose With probability kpk l 7p exactly k 7 1 supporters of A will vote and the candidates will tie The rest of the time B still wins because few A supporters end up voting This gives us Ubno votelvoter kpk 117p 2 1 7 pk kpk 117p Algebra con rms that the expected utility of voting is higher What about the m 7 k non voters Voting guarantees a win yielding 2 c Not voting results in a tie if all A supporters vote otherwise a win Thus UB no votelnon voter pk217pk Algebra con rms that this person would prefer not to vote Now lets think about what this model is telling us The probability that an A supporter votes is p 0 which is also the expected turnout in percentage terms of A supporters Differentiation or inspection reveals that as 0 increases individual voting likelihood and voter turnout decrease because voting is more expensive This makes sense How does a tight race affect voter turnout The closer the number of A supporters is to the number of B supporters ie the smaller m 7 k the more likely the A supporters are to all turn out and force a tie This ts a broadly observed pattern in US elections Public Goods and Reporting Crime In Chapter 8 you solved a model of a partnership using iterated dominance We begin this section by analyzing a similar model using Nash equilibrium Two employees work together in a team Worker 239 for 239 12 chooses effort 6 gt 0 to contribute to the team output y 61 62 in utility terms The output is divided evenly between the workers and each worker experiences a disutility of 6 From a social planner7s perspective the total welfare is maximized if the employees choose 61 62 1 Verify this by maximizing the the social welfare function 61 62 7 e 63 However if each employee takes the others effort as given she has the best response function e E which does not depend on the others7 behavior This is below the social optimum and each player is individually worse off compared to e 1 What does this remind you of Once again here is a game that resembles the Prisoners7 Dilemma It also feels like a public goods problem where each person free rides The com mon theme among these situations is that players do not enjoy the full social bene t of their actions If they exert more effort they generate additional surplus but they get only half of it The incentive to exert effort is muted and because the marginal cost of exerting effort increases with effort due to convex effort costs workers will stop earlier Exerting effort carries a positive externality for the other employee but because neither internalizes the others7 bene t effort is underprovided We see similar things happening with negative externalities as well Polluters do not bear the brunt of the social costs imposed by their actions so they overpollute These really are the same situations The refraining from the behavior with a positive externality imposes a negative externality and refraining from the behavior with the negative externality effec tively has a positive externality The team problem the free rider problem and the tragedy of the commons77 are manifestations of this same problem Next we apply Nash equilibrium to a model of reporting a crime A crime is observed by 71 people Each person wants the police to know about the crime but prefers someone else to tell the police The value of the police being informed is 1 but the cost of doing the informing is c where 1 gt c gt 0 You may already see where this is going private cost but everybody bene ts7surely a recipe for free riding Maybe so but how does behavior depend upon the number of people in the group With a larger group the individual cost of calling the police remains the same but doing so would make more people happier On the other hand in this simple model the individual decision maker does not incorporate the well being of others into her utility What about the poor victim ofthe crime ls he more likely to get help ifthere are more bystanders Let7s formalize the game and apply Nash equilibrium 0 There are 71 players 0 Each player chooses from Call Don7t call 0 Each player get utility 0 if no one calls 1 7 c if she calls and 1 if at least one person calls but she does not This game is a variant of a public goods problem in which k people are needed to provide the public good but in this case k 1 Are there any pure strategy Nash equilibria In fact there are n of them in each one exactly one person calls If that person switches she loses 11 7 c If anyone else decides to call as well they waste their phone call and lose 0 These Econ 171 Fall 2008 Lecture 11 October 30 In class game variant of B08 W incomplete information Consider a variant of B08 in which Player 1 is unsure whether Player 2 wants to go out with her or avoid her and thinks that these two possibilities are equally likely We could imagine that this assessment comes from Player l s experience half of the time she is involved in this situation she faces a player who wants to go out with her and half of the time she faces someone who wants to avoid her As in the standard game Player 2 knows Player l s preferences So Player 1 thinks that with probability 12she is playing the game on the left of Figure l and with probability 12she is playing the game on the right 0 B 2 wishes to meet 2 wishes to avoid Figure l Variant of ROS with incomplete information Player I is unsure whether 2 wants to meet her or avoid her thinking each possibility equally likely Player 2 knows which is the case Introduction to Bayesian Games and Incomplete Information 21 00 00 12 Figure 3 Sequential BoS De nition 1 When some players do not know the payo s of the others a game is said to have incomplete information Another term that is used to describe these games is Bayesian game
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