Nash+equilibrium+and+SPNE

Nash+equilibrium+and+SPNE - Nash equilibrium and subgame...

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Nash equilibrium and subgame perfect Nash equilibrium in normal and extensive for games An example solved in detail
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Three techniques illustrated by this example 1. How to use backward induction to solve an extensive form game for a subgame perfect Nash equilibrium. 2. How to construct the normal or strategic form from the extensive form (or game tree) representation of the game. 3. How to find the (pure-strategy) Nash equilibria in a normal form game by locating strategy pairs for the players that are best replies to one another. (This technique applies to all normal form games, not just ones derived from extensive form games.)
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Use backward induction to find subgame perfect Nash equilibrium To apply backward induction, we first look at the final decision nodes in the tree, that is, nodes after which there are no further actions to be chosen by any player. This game has two such nodes, the one at which 2 chooses between Top and Bottom and the one at which 1 chooses between Air and Ground . What would be the best action for each player if play were to reach either of these nodes?
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2’s choice between Top and Bottom leads to a payoff of 0 or 5 , respectively. Bottom is the optimal action.
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We record 2’s choice of Bottom with an arrow in the game tree.
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1’s choice between Air and Ground leads to a payoff of 1 or 7 , respectively. Ground is the optimal action.
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1’s choice of Ground gets an arrow in the game tree.
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Working backward up the tree, we analyze 2’s choice between High and Low by assuming that 2 anticipates that 1 will behave rationally if 1 is given the move to choose between Air and Ground . We have already seen that 1’s best action in that instance is to choose Ground . Thus 2’s choice is between High , which leads to a terminal node in which 2’s payoff is 4 , and Low , which in turn will induce a choice of Ground by 1, resulting in a payoff of 9 to player 2. Low is 2’s optimal action.
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A new arrow records 2’s choice of Low . 2’s choice is based not only on his or her own payoffs, but also 2’s belief that 1 would rationally pursue his or her own interests if given the choice of Air v . Ground . We are thus also assuming that 2 knows 1’s payoffs .
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Finally, we analyze 1’s choice between Up and Down by supposing again that both
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This note was uploaded on 11/10/2011 for the course POLI SCI 790:103 taught by Professor Blair during the Fall '09 term at Rutgers.

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Nash+equilibrium+and+SPNE - Nash equilibrium and subgame...

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