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1.2 Strategic Interactions Game Theory → a Game Plan, a Specification Of R.E.Marks 1998 Lecture 2-1 R.E.Marks 1998 Lecture 2-2 Game theory provides links: many diverse 1.2 Strategic Interactions situations have the same essential structure. Ðaprocurement manager trying to induce a Game theory → a game plan, a speci®cation of subcontractor to search for cost-reducing actions covering all possible eventualities innovations Ðanentrepreneur negotiating a royalty Strategic situations: arrangement with a manufacturing ®rm to ªin¯uence, to outguess, license the use of a new technology or to adapt to the decisions or lines of behaviour that others have just adopted or are expected to Ðasales manager devising a adoptº Ð Schelling. commission±payments scheme to motivate salespeople Look forward and reason backwards. Ðaproduction manager deciding between piece- rate and wage payments to workers a ®rm's Game theory is the study of rational behaviour in compensation committee designing a situations involving interdependence. managerial incentive system · May involve common interests: coordination Ð a manager's decision on how low to bid for a · May involve competing interests: rivalry government contract · Rational behaviour: players do the best they Ð an antique collector's decision on how high to can, in their eyes; bid in an auction · Because of the players' interdependence, a Ðatakeover raider's decision on what price to rational decision in a game must be based on a offer for a ®rm prediction of others' responses. Ðanegotiation between a corporation and a By putting yourself in the other's shoes and foreign government over the setting up of a predicting what action the other person will manufacturing plant choose, you can decide your own best action. Ð the haggling between a buyer and seller of a used car Ð collective bargaining between a trade union/employees and an employer R.E.Marks 1998 Lecture 2-3 R.E.Marks 1998 Lecture 2-4 1.2.1 Auctioning a Dollar 1.2.2 Schelling's Game (See O'Neal in the Package.) Rules: Rules: · First bid: 5¢ · single play, $4 to play · Lowest step in bidding: 5¢ · vote ªCº (CooÈperate) or ªDº (Defect). · Auction lasts until the clock starts ringing. · sign your ballot. (and commit to pay the entry fee.) · Highest bidder pays bid and gets $1 in return. · If x% vote ªCº and (100 ± x)% vote ªDº: · Second-highest bidder also pays, but gets x nothing. · then ªCºs' payoff = ( ____ × $6) − $4 100 · then ªDºs' payoff = ªCº payoff + $2 Write down the situation as seen by 1. the high bidder, and · Or: You needn't play at all. 2. the second highest bidder. WHAT HAPPENED? What happened? · numbers & payoffs. Escalation and entrapment · previous years? Examples? B cooÈperate for the common good or Dilemma: C D defect for oneself Public/private information Examples? R.E.Marks 1998 Lecture 2-5 R.E.Marks 1998 Lecture 2-6 Schelling's Game 1.2.3 The Ice-Cream Sellers (See Schelling in Package.) (See Marks in the Package) L C R 8 ªDº Ã Ã Ã 6 ªCº · Demonstration · Payoff matrix $ per participant · Incentives for movement? 4 · Examples? We can model this interaction with a simpli®cation: each ®rm can either: 2 · move to the centre of the beach (M), or · not move (stay put) (NM). 0 The share of ice-creams each sells (to the total 0 20 40 60 80 100 population of 80 sunbathers) depends on its move and that of its rival. Percentage of participants voting C Since each has two choices for its location, there Payoffs per participant are 2 × 2 = 4 possibilities. We use arrows and a payoff matrix1, which clearly outlines the possible actions of each and Note: the game costs $4 to join. the resulting outcomes. R.E.Marks 1998 Lecture 2-7 R.E.Marks 1998 Lecture 2-8 What are the sales if neither moves (or both NM)? Each sells to half the beach. The Ice-Cream Sellers What are the sales if You move to the centre (M) The other seller and your rival stays put at the three-quarter point? MNM What if you both move? ______________________ Given the analysis, what should you do? L L L M L 40, 40 L 50, 30 L L L L Does the model imply necessary reversion to the ______________________ You L L L mean? L L L No. (See the Marks in the Package.) NML 30, 50L 40, 40 L ______________________L L L TABLE 1. The payoff matrix (You, Other) A non-cooperative, zero-sum game, with a dominant strategy, or dominant move. _________ 1. See the Glossary in the Package for new meanings. R.E.Marks 1998 Lecture 2-9 R.E.Marks 1998 Lecture 2-10 1.2.4 The Prisoner's Dilemma The Prisoner's Dilemma (See Marks in the Package) The other player The Payoff Matrix: CD · The Cheater's Reward = 5 · The Sucker's Payoff = 0 ______________________ · Mutual defection = 2 L L L CL 4, 4L 0, 5 L · Mutual cooÈperation = 4 L L L ______________________ You L L L These are chosen so that: L L L 5+0<4+4 D5,0L L 2, 2 L ______________________L L L so that C,C is ef®cient in a repeated game. Need for communication TABLE 2. The payoff bimatrix (You, Other) cooÈrdination trust or? A non-cooperative, positive-sum game, Ef®cient Outcome: there is no other with a dominant strategy. combination of actions or strategies that would make at least one player better off without making Ef®cient at _____ any other player worse off. Nash Equilibrium at _____ Try your hand at playing the Prisoner's Dilemma over the 'Net: (See above for a de®nition of Nash Equilibrium.) http://serendip.brynmawr.edu/∼ann/pd.html R.E.Marks 1998 Lecture 2-11 R.E.Marks 1998 Lecture 2-12 1.2.5 The Capacity Game At a Nash equilibrium, each player is doing the best it can, given the strategies of the other Two ®rms each produce identical products and players. each must decide whether to Expand (E) its We can use arrows in the payoff matrix to see capacity in the next year or not (DNE). what each player should do, given the other A larger capacity will increase its share of the player's action. market, but at a lower price. The Nash equilibrium is a self-reinforcing focal The simultaneous capacity game between Alpha point, and expectations of the other's behaviour and Beta can be written as a payoff matrix. are ful®lled. The Nash equilibrium is not necessarily ef®cient. The Capacity Game The game above is an example of the Prisoner's Beta Dilemma: in its one-shot version there is a con¯ict between collective interest and self-interest. ______________________DNE Expand L L L DNE $18, $18 $15, $20 Example ______________________L L L Alpha L L L Given the social costs associated with litigation, L L L why is it increasing? Expand______________________L $20, $15 L $16, $16 L David Ogilvy has said, ªHalf the money spent on advertising is wasted; the problem is identifying TABLE 3. The payoff matrix (Alpha, Beta) which half.º Is the explanation for the amount of advertising a A non-cooperative, positive-sum game, Prisoner's Dilemma? with a dominant strategy. Ef®cient at _____ Nash Equilibrium at _____ R.E.Marks 1998 Lecture 2-13 R.E.Marks 1998 Lecture 2-14 1.3 Modelling Players' Preferences Simple Games: · The Dollar Auction In two-person games, each of the two players has · The n-person Prisoner's Dilemma only n possible actions: · The Ice Cream Sellers · The Prisoner's Dilemma ∴ represent the game with a n × n payoff matrix. · The Battle of the Bismark Sea · Chicken! Two actions per player: n =2. · The Boxed Pigs ∴ Each player faces four possible combinations. · The Battle of the Sexes · The Gift of the Magi For a one-shot game in pure strategies · Bargaining (i.e., no dice rolling or mixing of pure strategies): · Vickrey auction need only rank the four combinations: Further games, from Gardiner (pp.): best, good, bad, worst: · Battle of the Networks: 38±41 · Video System Coordination (2P & 3P): 50±51, → payoffs of 4, 3, 2, and 1, respectively 99±100 Larger numbers of possible actions: · Cigarette Advertising on TV: 51±53 · Matching Pennies: 68±69 harder to rank the larger number of outcomes · Market Niche (Symm. & Asymm, 2P & 3P): (with three actions there are 3 × 3 = 9), 69±70,79±81,97±99 but ranking suf®cient. · Everyday Low Pricing Ð Sears: 81±84 (i.e. ordinal preferences, instead of asking ªby how · 3P Competitive Advantage: 96±97 much is one outcome preferred to another?º) · Stonewalling Watergate: 102±104 · Tragedy of the Commons: 108±111 Later, with mixed strategies (probabilistic or · Cournot Competition: 124±127 dice-throwing) and unpredictability: · Bertrand Market Game: 129±133 · use probabilities over actions and · Telex versus IBM: 149±151 · Conscription: Reluctant Volunteers: 157±159 · the expected values of the possible outcome · Mutually Assured Destruction Ð MAD: · use cardinal measures over the amounts, 160±164 usually dollar amounts, which are · Hawk versus Dove: 213±215 unambiguous, and the numbers matter! · Caveat Emptor (Buyer Beware): 240±242 · Depositor versus Savings & Loan: 286±288 R.E.Marks 1998 Lecture 2-15 R.E.Marks 1998 Lecture 2-16 · Voting Game: 440±443 1.4 Concepts and Tools The following concepts & tools are introduced: The Ice-Cream Sellers: payoff matrix incentives to change Ð use arrows! dominant strategy The Prisoner's Dilemma: possibility of repetition ef®ciency Ð Pareto Optimality bimatrix of payoffs Ð both players' non-zero-sum game inef®cient equilibria The Battle of the Bismark Sea: zero-sum game iterated dominant strategy Nash equilibrium Boxed Pigs: rationality game in normal form (payoff matrix) weakness may be strength ef®cient equilibria game in extensive form (game tree) information set The Battle of the Sexes: coordination, not rivalry ®rst-mover advantage focal points See the De®nitions in Lecture 6 (from Rasmusen), and the Glossary in the Package (from Baird Gertner & Picker)..
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