Econ104_Game Theory Tools

Econ104_Game Theory Tools - Game Theory Tools Appendix A:...

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Game Theory Tools Appendix A: Normal-Form Games Appendix B: Extensive-Form Games Appendix C: Undercut-Proof Equilibria
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Game Theory: Introduction Game Theory formalizes strategic situations e.g. situations in which one firm’s profits depend on what other firms do – with the goal of deriving the best strategy for each participant and the resulting equilibrium. Applied in many areas of economics and other social sciences. Most useful when the number of participants is small and when they have a notable effect on each other’s outcomes.
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Normal-form games Also called simultaneous games . “Players” choose “actions” which affect their own and all other agents’ “payoffs”. Everybody chooses his/her action at the same time (i.e. simultaneously ). Players are individuals or firms who can take decisions. Payoff is the utility or profit of the player. Action (=strategy) is any decision variable for the player, e.g. price or quantity set by a firm.
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More formal definition: A normal form game is described by a set of players a set of actions for each player, i.e. a list of possible decisions a payoff function for each player, which tells us the player’s utility/profit for each combination of actions taken by all players. Game theorists use payoff tables to summarize the parameters of simultaneous games. The payoff table contains all possible actions for each player and the payoffs associated with each action, taking into account the other players’ actions.
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Analyzing payoff tables: Strategy ‘Left’ Strategy ‘Right’ Strategy ‘Up’ ( Π 1,UL , Π 2,UL ) ( Π 1,UR , Π 2,UR ) Strategy ‘Down’ ( Π 1,DL , Π 2,DL ) ( Π 1,DR , Π 2,DR ) Player 1’s strategies Player 2’s strategies
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Example: Firm 2 Low price High price Firm 1 Low price 100,100 300,0 High price 0, 300 200, 200
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Example continued Firm 1 and firm 2 are the players. Each firm has the action set {low price, high price}. Both firms have the following payoff function: π (low, low) = 100 π (low, high) = 300 π (high, low) = 0 π (high, high) = 200 where the 1 st argument is the own action, the 2 nd argument is the other player’s action.
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Equilibrium concepts Which combination of actions get chosen? How do we know that any combination is a stable outcome? To answer this, we use the concept of “equilibrium”. This is familiar from simple supply and demand analysis: The market equilibrium is the price-quantity combination for which supply equals demand. We expect that we will observe the price-quantity combination for which the market is in equilibrium. Just as market equilibrium is the economist’s favorite tool in supply and demand analysis, Nash equilibrium is the game theorist’s favorite tool for predicting the outcome in any kind of game.
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Nash Equilibrium Nash equilibrium is defined as a combination of players’ strategies (actions) such that no player finds it beneficial to deviate from its
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Econ104_Game Theory Tools - Game Theory Tools Appendix A:...

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