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© Malcolm Brady, 2011 Next Generation Management Interdependent decision making: introduction to game theory Malcolm Brady DCU Business School Dublin City University MSBM/MECB/MMK 28 th March 2012
© Malcolm Brady, 2011 Game theory Orginally developed in the early 20 th century by mathematicians and economists –von Neumann, Morgenstern to address the problem of interdependent decision making –ie. a situation where a decision made by one party influences a decision made by another party and has since become the dominant paradigm in industrial organisation economics, –ousting the structure-conduct-performance paradigm and is steadily making inroads into the field of strategic management Eight recent Nobel prizewinners have been game theorists – 1994:Nash, Selten, Harsanyi; 2005: Aumann, Schelling; 2007: Myerson, Hurwicz, Maskin. Examples: Dark Knight Beautiful Mind The Good…
© Malcolm Brady, 2011 ‘It is not from the benevolence of the butcher, the brewer, or the baker that we expect our dinner, but from their regard to their own self-interest …and he is in this, as in many other cases, led by an invisible hand to promote an end which was no part of his intention’ Adam Smith, 1776 (1994, I.15, IV.485) Rational behaviour The world is driven by self-interest Rational economic man – maximises their own utility Smith saw price and the market as the invisible hand Individual action may lead to an unexpected end
© Malcolm Brady, 2011 Menu Starter: salad $4 or prawns $10 Mains: burger $13 or steak $25 Dessert: icecream $4 or pavlova $4 Coffee: $3 Wine: bottle $25 shared do you drink: a little or a lot
© Malcolm Brady, 2011 ‘Two roads diverged in a wood, and I I took the road less travelled by, And that has made all the difference.’ Robert Frost Yellow wood Road less travelled Road more travelled Made all the difference Dixit and Nalebuff, 1993:34 A simple decision
© Malcolm Brady, 2011 Newcleaners enter do not enter Fastcleaners accomodate fight price war $0 to Newcleaners -$200k to Newcleaners $100k to Newcleaners Expected value of entering =.5x$100k +.5x(-$200k) = - $50k Expected value of not entering = 0 => Do not enter How to determine probabilities of what Fastcleaners will do? Dixit and Nalebuff, 1993:37 Decision Tree
© Malcolm Brady, 2011 Newcleaners enter do not enter Fastcleaners accomodate fight price war $0 to Newcleaners $300k to Fastcleaners -$200k to Newcleaners -$100k to Fastcleaners $100k to Newcleaners $100k to Fastcleaners Dixit and Nalebuff, 1993:37 Now what happens? Look ahead, and reason backwards. Game tree
© Malcolm Brady, 2011 Form of games Extensive form –Structured as a decision tree –Multiple stages are evident –Becomes cumbersome if games are complex Strategic form (or normal form) –Structured as a payoff matrix –Summarised form
© Malcolm Brady, 2011 Extensive form Player –Each independent decision maker is a player –Chance can also be a player (eg. throw of a dice, act of God) Move –A decision point in a game, at which alternative choices are available to a player Action –The alternative choices available to a player at a move Outcome –An end point of the game Payoff –The value received by a player at an outcome Game tree –The order of decision making is displayed as a tree –A tree is a connected graph containing no loops and beginning at a single node (known as the root node) –A tree is not necessarily a unique representation of a game
© Malcolm Brady, 2011 Assumptions Each player acts so as to maximise their utility ie. to gain maximum payoff Each player has preferences over the various outcomes of the game and chooses actions to achieve their preferred outcome –ie. each player is ‘rational’ in the sense that, given two alternatives, he will always choose the one that he prefers ie. the one with the larger utility Each player has full knowledge of the game in extensive form –Each player knows the preference pattern and payoff function of the other players –Each player is fully aware of the rules of the game
© Malcolm Brady, 2011 Strategic (normal) form In principle we could ask each player to state in advance what he/she would do in each situation which might arise in the play of a game. From this information an umpire could carry out the play of the game without further aid from the players and thereby determine the payoffs. Such a description of decisions for each possible situation is known as a pure strategy for a player. If player i has q information sets then q-tuples of integers (y 1, y 2,…y q ) represent pure strategies We can then label the strategies 1 to t where t is the total number of strategies available to the player. Note that t is a finite number but could be a large number. We can now represent the game using only the strategies for each player.
© Malcolm Brady, 2011 Strategic form – payoff matrix Player 1 Player 2 A ba 1,1 2,21,2 2,1 A,B: strategies available to player 1 a,b: strategies available to player 2 (x, y): payoff for players 1, 2 respectively B A B a b b a (1, 1) (2, 2) (1, 2) (2, 1) The two representations are equivalent
© Malcolm Brady, 2011 Example: advertising campaign Dunnes Tesco Ireland small bigsmall $100m big Adapted from Grant, CSA, 2010:101 $100m $70m $110m $70m $80m See also: Brandenburger and Nalebuff,1995, HBR, July-Aug
© Malcolm Brady, 2011 Strategic form This strategic form game is equivalent to the extended form (tree structure) above In this example the two player’s strategies consist of just one action; however, a strategy is a set of actions and the set may comprise of more than one action Formally, a game in normal form consists of –A set of n players –n sets of pure strategies s i, one for each player –n linear payoff functions m i, one for each player, whose values depend on the strategy choices of all the players.
© Malcolm Brady, 2011 Zero Sum Games In zero sum games we have two players and each player has only one move. These moves are taken simultaneously (or if in succession the player who moves second has no knowledge of the choice made by the first player) This is not as restrictive as it seems as many interesting situations involve only two players and all extensive form games can be transposed into normal form where each player has only one move (ie. makes one choice from a set of strategies). Where players have strictly opposing preferences among outcomes then it is a strictly competitive game eg. player 1 prefers outcome x to outcome y and player 2 prefers outcome y to outcome x Payoffs for players are diametrically opposed ie. p 2 = - p 1 or p 2 + p 1 = 0 (zero sum)
© Malcolm Brady, 2011 Zero Sum Game example α1α1 α2α2 α3α3 α4α4 β4β4β3β3β2β2β1β α5α5 Luce and Raiffa, 1957:61
© Malcolm Brady, 2011 Infinite loop Player 1 would like to play strategy α4 and get a payoff of 25; but on realising this then player 2 would play β3 and keep her loss down to 2; player 1, realising this, would then prefer to play α2 and so gain 8; but then player 2 would prefer to play β1 and lose zero; player 1 would then prefer α1 and gain 16; player 2 would then prefer β3, player 1 α2, then player 2 β1 … We could equally start with player 2 and get a different infinite sequence of choices
© Malcolm Brady, 2011 Maximin and Minimax Player 1: determines the minimum he can gain by playing each strategy, then chooses the maximum of these minima Player 2: determines the maximum she can lose by playing each strategy, then chooses the minimum of these maxima => Player 1 plays α3 and thereby ensures that he receives a payoff of at least 4 =>Player 2 plays β2 and ensures that she makes a loss of no more than 4
© Malcolm Brady, 2011 Maximin α1α1 α2α2 α3α3 α4α4 β4β4β3β3β2β2β1β α5α5
© Malcolm Brady, 2011 Minimax α1α1 α2α2 α3α3 α4α4 β4β4β3β3β2β2β1β α5α5
© Malcolm Brady, 2011 Equilibrium pair A pair of strategies (α, β) is an equilibrium pair if the corresponding entry in the payoff matrix is the minimum value in its row and the maximum value in its column At that cell neither player will have an incentive to change his strategy Existence: strictly competitive games do not necessarily have an equilibrium pair Uniqueness: several equilibrium pairs may exist in a strictly competitive game Is (α3, β2) above an equilibrium pair?
© Malcolm Brady, 2011 Outcome represents number of bombing days by US planes on Japanese convoy Both commanders chose north channel Japanese convoy sighted after one day and suffered severe losses Cannot say that Japanese commanded erred in his decision. The north route was at least as good as the south route against either of the two US strategies Example: Bismarck Sea New Britain J US N: Poor visibility S: Clear weather N N S S US J Source: Luce and Raiffa, 1957:64
© Malcolm Brady, 2011 Minimax theorem For any two person zero sum game there exists a mixed strategy (maximin) for player 1 which guarantees him at least v, and a mixed strategy (minimax) for player 2 which guarantees that player 1 gets at most v. These mixed strategies are in equilibrium. The unique number v is called the value of the game
© Malcolm Brady, 2011 Non-zero sum games We now move on to two person non-zero sum non-cooperative games By a cooperative game is meant a game in which the players have complete freedom of preplay communication to make joint binding agreements In a non-cooperative game no preplay communication is permitted Cooperation does not arise in zero sum games because players interests are diametrically opposed β2β2β1β1 a 11 b 11 α1α1 α2α2 a 12 b 12 a 21 b 21 a 2,2 b 2,2
© Malcolm Brady, 2011 Battle of the sexes Couple going out Man prefers thriller to romance Woman prefers romance to thriller Both prefer to agree on something and go together than go out separately thriller romancethriller romance 2,1 1,20,0 man woman Video explaining mixed strategy NE for this game:
© Malcolm Brady, 2011 Coordination game BoS is an example of a coordination game –Issue is ‘how to achieve coordination’? Shows power of declaring your strategy and sticking to it –Eg. man declares he has bought cinema tickets and, guess what, it’s for the latest thriller –A credible pre-play commitment In a zero sum game declaring your strategy in advance is never to your advantage In a coordination game it can be to your advantage to declare in advance and have a reputation for inflexibility –Second player acting in their own self-interest works to the advantage of the first player
© Malcolm Brady, 2011 Dominant strategy Solution is found by the method of iterated dominance For player II –R strictly dominates M –ie. a iR >a iM, i=U,M,D Then for player I –U strictly dominates M –U strictly dominates D Then player II prefers L to R => play is (U, L) and outcome is (4, 3) 4,3 5,16,2 3,0 2,1 9,6 8,4 2,8 3,6 U D M RML Fudenberg and Tirole, 1991:5
© Malcolm Brady, 2011 Iterated dominance Not all games can be solved by iterated dominance Order of iteration does not affect the prediction under strict dominance –But does under weak dominance Result also depends on behaviour and anticipated behaviour of players –Eg. iterated dominance gives (U,L) as solution but experiments show that D is often chosen by students Because of fear of other player making ‘mistake’ ie. not behaving rationally and first player ending up with ,66,5 -100,9 U D RL 8,10 Fudenberg and Tirole, 1991:8 7,16
© Malcolm Brady, 2011 Prisoner’s Dilemma Iterated dominance predicts (testify, testify) with an outcome of (-3,-3) ie. both prisoners going down for 3 years Even though a solution exists (don’t, don’t) which is better for both players Self interest can lead to inefficient outcomes Even if players agree not to testify beforehand the agreement will not be binding: each will be motivated to defect from the agreement testify don’t testify don’t testify -1,-1 -3,-30,-5 -5,0 We must re-consider the concept of optimality when we are dealing with several independent decision makers and where outcomes are interdependent See video explaining the prisoner’s dilemma:
© Malcolm Brady, 2011 Teamwork Work leads to a contribution of 1 Shirk leads to a contribution of zero Team output is 4 *sum of contributions Output is equally divided Cost of working is 3;cost of shirking is zero Moral hazard in teamwork –Is there is a motivation to shirk? 2,-10,0 -1,2 work shirk work 1,1 Fudenberg and Tirole, 1991:10
© Malcolm Brady, 2011 Price and profit Each firm has to choose whether to set a low or a high price Both can do well if both set high prices Firm gains advantage over rival if she sets high price while you set low Both firms are driven to the prisoner’s dilemma outcome 4,12,2 1,4hi lo hi 3,3 McAleese, 2004:145
© Malcolm Brady, 2011 Nash equilibrium ‘An array of strategies, one for each player, such that no player has an incentive (in terms of improving his payoff) to deviate from his part of the strategy array’ (Kreps 1990:28) For (α1,β1) to be a Nash Equilibrium in pure strategies a 11 >a 21 and b 11 >b 12 If both players settle on this pair of strategies then neither will have an incentive to move from their choice – neither player can improve their payoff by shifting to an alternative strategy Nash’s theorem – all non-cooperative games with a finite number of players and strategies have a mixed strategy equilibrium β2β2β1β1 a 11 b 11 α1α1 α2α2 a 12 b 12 a 21 b 21 a 2,2 b 2,2
© Malcolm Brady, 2011 Examples of Nash Equilibria 0,01,2 0,02,1 10,04,4 0,108,8 Battle of the Sexes: two NE in pure strategies exist Prisoners dilemma: one NE in pure strategies exists 10,04,4 5,78,8 Payoffs in top right cell altered: no NE in pure strategies exist 3,03,3 0,35,5 Stag hunt: two NE in pure strategies exist
© Malcolm Brady, 2011 Focal point Multiple Nash equilibria can exist in a game. This raises the issue of which NE players will settle on. Coordination problem can be solved by focal consideration. Both drivers naturally turn left, and both know that the other driver will naturally turn left and so (left, left) is the likely equilibrium But, if both drivers are foreigners and this is obvious (eg. both cars have foreign registrations) then (right,right) also becomes a possibility. Outcome here is less certain as there are now two focal considerations Two cars round a corner both of them in the centre of a narrow road l r r l -2,-2 -3,-30,0 -3,-3r l lr Two NE equilibria exist (l,l) and (r,r). (l,l) is Pareto dominant as payoff is better for both players ie. (0,0) is better than (-2,-2) for both NE
© Malcolm Brady, 2011 Stag Hunt game If hunters cooperate they get a stag If hunters hunt separately they each get a hare (Stag, stag) is pareto optimal Payoffs are best for at least one player and at least as good for other players But (hare, hare) is risk dominant –Lower ultimate payoff but less risky has a higher security level for each player 3,03,3 0,35,5 stag hare Party game: replace stag with ‘arrive early’ and hare with ‘arrive late’ If a group of hunters set out to take a stag, they are fully aware that they would all have to remain faithfully at their posts in order to succeed; but if a hare happens to pass one of them, there can be no doubt that he pursued it without qualm, and that once he had caught his prey, he cared very little whether or not he had made his companions miss theirs. Jean-Jacques Rousseau, quoted in Fudenberg and Tirole 1991
© Malcolm Brady, 2011 Chicken Similar to Battle of the Sexes But is an anti-coordination game With disastrous consequences for coordination on (tough, tough) 1,2 0,0 2,1-1,-1 tough weak Logan Airport Near Miss 9 June 2005 Aer Lingus A330 and US Airways B737 planes take off on two different runways that intersect See simulation See report What factors saved the day here? Aer Lingus Airbus A330 US Airways Boeing 737 Tower
© Malcolm Brady, 2011 Repeated Game: Tit for Tat Axelrod set up computer tournament where Tit for Tat strategy yielded best outcome in an infinitely repeated game Tit for Tat: cooperate in the first move and thereafter do whatever the other player did on previous move => cooperation can emerge even in a world of egoists without central authority World without governance not necessarily one where life is ‘solitary, poor, nasty, brutish and short’ (Hobbes, 1651)! Axelrod, 1984:10
© Malcolm Brady, 2011 Tit for Tat Nice Always begin by cooperating; never first to defect Forgiving Only defect once, then resume cooperation Retaliatory If other player defects, always punish by defecting on next move Clear Other player quickly comes to understand tit for tat strategy Robust Tit for tat proved best strategy in Axelrod’s tournaments; also proved best strategy in John Maynard Smith’s evolutionary simulations (survival of the fittest rule) Tit for Tat is a trigger strategy Grim trigger is another trigger strategy, but unforgiving: cooperate initially but, on defection by opponent, defect forever
© Malcolm Brady, 2011 Tournament Two players playing tit for tat both achieve payoffs of 3+3δ+3δ 2 +3δ 3 +…=3/(1- δ) Suppose player 2 plays all defects. Then player 1 achieves 0+δ+δ 2 +δ 3 +…=δ /(1- δ) and player 2 achieves 5+δ+δ 2 +δ 3 …= 5+δ /(1- δ) Player 2 is better off playing tit for tat provided 3/(1- δ)>5+δ /(1- δ) ie. δ>½ Above was the model used in the tournaments δ is the discount factor 5,01,1 0,5coop defect coop 3,3
© Malcolm Brady, 2011 Repeated prisoner’s dilemma 3+3δ+3δ2+3δ3+…=3/(1- δ) 5+δ+δ2+δ3…= 5+δ/(1- δ) 0+δ+δ2+δ3+…=δ/(1- δ) Tit for tat All defect 1+δ+δ2+δ3+…=1/(1- δ) For discount rate of 0.9 this reduces to 30,30 14, 910,10 9,14 This game is not dominance solvable but we will see shortly that the pair of strategies (tit for tat, tit for tat) yield a Nash equilibrium with payoffs of (30,30) Note that the pair of strategies (all defect,all defect) with payoffs (10,10) is also a Nash equilibrium
© Malcolm Brady, 2011 Folk Theorem In an infinitely repeated game it is possible to obtain the cooperative result, unless interest rates are too high (or discount rates too low) Martin, 2001: Player 1 payoff Player 2 payoff Player 1 can ensure payoff of 1 Player 2 can ensure payoff of 1 Folk theorem says that any outcome in shaded area can be obtained in an infinitely repeated game Both players can do better than they would under a one-shot game
© Malcolm Brady, 2011 Backward Induction Start at last decision node Ask what choice would decision maker at that node make Use dotted line to indicate this decision Then move back a decision node and repeat the process Continue until you reach the root node The dotted line path indicates the solution found by backward induction Strategy pair (R, (rl)) is solution and yields payoff (2,1) Note that a player’s strategy requires a choice at each decision node for that player L R l r r l (3, 1) (0, 0) (2, 1) (1, 2)
© Malcolm Brady, 2011 Subgame Perfection Subgame perfection places a further restriction on NE and narrows down the possible solutions to a game Subgame perfection implies that each subgame must also represent a NE Two NE exist: (R,rl) and (L,rr) but only (R,rl) is subgame perfect ie. these choices would also made in every subgame (L,rr) is not SPNE because player two will not choose r in the subgame across L R ll lrrlrr 3,1 2,1 0,0 3,1 2,12,1 0,0 1,2 1,21,2 Red: highest in column Green: highest in row Highest in column and highest in row indicates NE Selten Gibbons, 1992:115 r l 2 (0, 0) (2, 1)
© Malcolm Brady, 2011 Credible Threat NE of (L,rr) is sustained by the threat by player 2 of playing r But this threat is not credible –Refer back to earlier slide where threat to fight new entrant with price war was not credible Easier to see this in extensive form; not obvious in normal form Technical note: A subgame is a subset of the game that is itself a game in extensive form. It starts at a singleton node, and if any information set is reached then all nodes in that information set must be reachable
© Malcolm Brady, 2011 Some further reading Brandenberger A, Nalebuff B ‘The right game: use game theory to shape strategy’, Harvard Business Review 73(4):57-71 Gibbons, R An introduction to applicable game theory. Journal of Economic Perspectives 11(1): Dixit, A and B Nalebuff Thinking strategically, WW Norton. Dutta, P Strategies and Games, MIT Press.
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