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ECONOMICS OF PRICING AND DECISION MAKING Lecture 1

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What makes a business successful? Providing a service that customers like Building partnerships Being ahead of competitors Building brand value ...“Interactions” with customers, suppliers, competitors, regulators, people within the firm...

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What is game theory? ...a collection of tools for predicting outcomes of a group of interacting agents ... a bag of analytical tools designed to help us understand the phenomena that we observe when decision makers interact (Osborne and Rubinstein) ...the study of mathematical models of conflict and cooperation between intelligent rational decision makers (Myerson)

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What is game theory? Study of interactions between parties (e.g. individuals, firms) Helps us understand situations in which decision makers interact: strategies & likely outcome Game theory consists of a series of models, often technical as well as intuitive The models predict how parties are likely to behave in certain situations

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The Game: Strategic Environment Players Everyone who has an effect on your earnings (payoff) Actions: Choices available to the players Strategies Define a plan of action for every contingency Payoffs Numbers associated with each outcome Reflect the interests of the players

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Strategic Thinking Example: Apple vs. Samsung Apple’s action depends on how Apple predicts Samsung’s action. Apple’s action depends on how Apple predicts how Samsung predicts the Apple’s action. Apple’s action depends on how Apple predicts how Samsung predicts how Apple predicts the Samsung’s action. etc…

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The Assumptions Rationality Players aim to maximize their payoffs, and are self-interested. Players are perfect calculators Players consider the responses/reactions of other players Common Knowledge Each player knows the rules of the game Each player knows that each player knows the rules Each player knows that each player knows that each player knows the rules Each player knows that each player knows that each player knows that each player knows the rules Each player knows that each player knows that each player knows that each player knows that each player knows the rules ...

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History of game theory 1928, 1944: John von Neumann 1950: John Nash 1960s: Game theory used to simulate thermonuclear war between the USA and the USSR 1970s: Oligopoly theory 1980s: Game theory used Evolutionary biology Political science More recent applications: Philosophy, computer science 1994, 2005, 2007, 2012: Economics Nobel prize

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Lectures 1-3: Simultaneous games Nash equilibrium Oligopoly Mixed strategies 4-5: Sequential games Subgame perfect equilibrium Bargaining 6: Repeated games Two firms interacting repeatedly

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Lectures 7: Evolutionary games How do players “learn” to play the Nash equilibrium 8-9: Incomplete information Cooperation and coordination with incomplete information Signaling, and moral hazard. 10: Auctions Strategies for bidders and sellers

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Assessment Assessment consist is a final exam: 100% exam 2-hour Section A: 5 compulsory questions, at most 3 "mathematical/analytical" questions. (10 marks each) Section B: choose 1 essay question from a list of 2. (50 marks)

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SIMULTANEOUS GAMES WITH DISCRETE CHOICES Pure strategy Nash Equilibrium

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Simultaneous games with discrete choices A game is simultaneous when players choose their actions at the same time or, choose their actions in isolation, without knowing what the other players do Discrete choices: the set of possible actions is finite e.g. {yes,no}; {a,b,c}. Opposite of continuous choices: e.g. choose any number between 0 and 1.

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Strategic Interaction Players: Reynolds and Philip Morris Payoffs: Companies’ profits Strategies: Advertise or Not Advertise Strategic Landscape: Each firm initially earns $50 million from its existing customers Advertising costs a firm $20 million Advertising captures $30 million from competitor Simultaneous game with discrete choices

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Representing a Game (strategic form / normal form) What is the likely outcome? We want a “stable”, “rational” outcome.

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Solving the game: Nash equilibrium The Nash equilibrium, is a set of strategies, one for each player, such that no player has incentive to unilaterally change his action The NE describes a stable situation. Nash equilibrium: likely outcome of the game when players are rational Each player is playing his/her best strategy given the strategy choices of all other players No player has an incentive to change his or her action unilaterally

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Solving the Game Can (No Ad,No Ad) be a Nash equilibrium? No, 60>50 Can (No Ad,Ad) be a Nash equilibrium? No: 30>20 Can (Ad,No Ad) be a Nash equilibrium? No: 30>20

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Solving the Game Can (Ad,Ad) be a Nash equilibrium? YES: 30>20 If Philip Morris “believes” that Reynolds will choose Ad, it will also choose Ad. If Reynolds “believes” that Philip Morris will choose Ad, it will also choose Ad. (Ad, Ad) is a “stable” outcome, neither player will want to change action unilaterally.

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Equilibrium vs. optimal outcome The optimal outcome is the one that maximizes the sum of all players’ payoffs. (No Ad, No Ad) The NE does not necessarily maximize total payoff. (Ad,Ad). The NE is individually rational, but not always collectively rational.

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Game of cooperation (prisoner’s dilemma)

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Nash equilibrium existence Q: Does a NE always exist? A: Yes (in almost every cases). [If there is no equilibrium with pure strategies, there will be one with mixed strategies.] Theorem (Nash, 1950) “There exists at least one Nash equilibrium in any finite games in which the numbers of players and strategies are both finite.”

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Nash equilibrium A formal definition Any social problem can be formalized as a “game,” consisting of three elements: Players: i=1,2,…,N i’s Strategy: i’s Payoff:

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Nash equilibrium A formal definition Definition: A Nash Equilibrium is a profile of strategies such that each player’s strategy is an optimal response to the other players strategies: If all players play according to the NE, no player has any incentive to change his action unilaterally. Why is the NE the most likely outcome: Any other outcome is not “stable”. In the long term, players learn how to play and always select the NE

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How to find the Nash equilibrium? There are two techniques to find the NE Successive elimination of dominated strategies Best response analysis

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Elimination of dominated strategies (1st method) Procedure: eliminate, one by one, the strategies that are strictly dominated by at least one other strategy. Consider two strategies, A and B. Strategy A strictly dominates Strategy B if the payoff of Strategy A is strictly higher than the payoff of Strategy B no matter what opposing players do. For Philip Morris, Ad dominates No Ad: π(Ad,any)> π(No Ad,any). For Reynolds Ad also dominates No Ad. Strictly dominated strategies can be eliminated, they would not be chosen by rational players.  No Ad can be eliminated for both players.

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Elimination of dominated strategies

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Elimination of dominated strategies The order in which strategies are eliminated does not matter. Select any player, any strategy, and check whether it is strictly dominated by any other strategy. If it is strictly dominated, eliminate it. When several strategies are strictly dominated, it does not matter which one you eliminate first.

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Elimination of dominated strategies

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Elimination of dominated strategies

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Weak dominance Strategy A weakly dominates strategy B if its strategy A’s payoff is in some cases higher (>) and in some cases equal () to strategy B’s payoff. Alternative scenario: One strategy weakly dominates the other 60>50 30=30

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Weak dominance Weakly dominated strategies cannot be eliminated. In some cases, when strategies are only weakly dominated, successive elimination can get eliminate some Nash equilibria.

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Best response analysis (2nd method)

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Best response analysis

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Exercise

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Comparing the two methods The two methods for finding the NE are NOT equivalent. The best response analysis is fully reliable, and always finds the NE. Sometimes, the elimination of dominated strategies will fail to find the NE. This may happen when that are more than one NE.

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Comparing the two methods Example of an entry game: Two businesses must choose which market to enter. This is a game of coordination (not cooperation!): class of games with multiple NE (two in this case).

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Comparing the two methods 1st method: The game is not dominance solvable, there are no dominated strategies. 2nd method: With best response analysis, both equilibria are found. When best-response analysis of a discrete strategy game does not find a Nash equilibrium, then the game has no equilibrium in pure strategies.