E. Game theory
Simultaneous games Prisoners’ dilemma
• 2 prisoners are being asked separately
• 2 prisoners have to confess/not confess to a crime Game mechanism
• If both confess,
they will be convicted & sentenced to 10 years in prison
• If both do not confess,
they will be charged with less offence & receive 3 years in prison
• If one confesses & other player does not confess, confessor receive 1-year in prison
non-confessor receives 25-year sentences
(Each suspect wants to betray other player because of the attractiveness of the reduced prison sentence)
Player A (row)
If Player B (column) chose confess, A picks confess (-10) If Player B (column) chose not confess, A picks confess (-1) Best response action = Player A picks confess
Player B (column)
If Player A (row) chose confess, B picks confess (-10) If Player A (row) chose not confess, B picks confess (-1) Best response action = Player B picks confess
Strictly dominant strategy
• Strategy that provides player with a strictly higher payoff than all other strategies
• Rational players will always pick strictly dominant strategy if it exists as best response
• When all players have strictly dominant strategy, equilibrium should contain strictly dominant strategies
• In prisoner’s dilemma game,
both prisoners have strictly dominant strategy => “confess”
so we predict equilibrium= both prisoners receive 10-year sentence Weakly dominant strategy
• Strategy that provides player with no lower payoff than all other strategies
• Rational players will always pick a weakly dominant strategy, if it exists as best response
• When all players have weakly dominant strategy, equilibrium should contain weakly dominant strategies
Battle of sexes 3x3 game
Husband & wife need to coordinate how to spend their day without observing each other
Dominated strategy NEVER PICK
• Strategy that provides player with lower payoff than any other strategy
• Rational players will never pick dominated strategy
• If strictly dominant strategy exists, all other strategies must be strictly dominated
• If weakly dominant strategy exists, all other strategies must be weakly dominated
• Even when there are no strictly/weakly dominant strategies, a player could still have dominated strategies
• In battle of the sexes,
o “Wedding” = dominated strategy because both player never played
o Given remaining possibilities,
“Opera” = weakly dominant strategy for both players hence, predicted equilibrium
Sometimes, eliminating dominated strategies may not be enough to determine the equilibrium
Tennis
• 2 players in a game of tennis
• To begin each point,
server chooses one of “Left, Centre, Right” and receiver prepares to return with one of “Left, Centre, Right”
Nash equilibrium
• A set of strategies such that, when all other players use these strategies, no player can obtain a higher payoff by choosing a different strategy
• If all players have dominant strategies, there is Nash equilibrium
• Dominated strategies can never be part of a Nash equilibrium
• Even when 1 (or more) player does not have a dominant strategy, Nash equilibrium could still exist
In tennis game,
• No player has a dominant strategy
• We can eliminate server’sdominated strategy “C” but this does not show any dominant strategy for either player given remaining possibilities
• While {L, C} are not dominant strategies, no player can do better by switching strategies
• So, {L, C} is a Nash equilibrium, thus predicted equilibrium How to use Nash equilibrium (NE) as a solution concept
• If all players have dominant strategies,
they cannot do better than playing these strategies => there’s NE
• When some players have dominant strategies & some not,
do the latter have best responses to formers’ dominant strategies? => If so, then NE
• When no player has a dominant strategy,
can we eliminate dominated strategies to find dominant strategies? If so, then NE
Can have multiple predicted outcomes in a game when we use Nash equilibrium
• No dominant strategy
• No dominated strategy
Look at intersection of best responses
All 3 are NE
Extra considerations (refinement)
• Psychology & role of fairness: make 50/50 a “focal point”
Sequential games
We could specify a new Normal Form to find all the NE
• The Normal Form approach has some issues: • The normal form doesn’t give us any sense of timing. We have a hard time reconstructing the original game. • A bigger issue is that some of the NE of the game are strange. • If P1 chooses T, P2 using strategy {V, V} will be getting 0. • She could do better if she changed her strategy after P1’s initial choice and switched to T,V • Thus, the Nash Equilibrium V, {V, V}
exists in part because P2 is not switching strategies when it is in his interest. • We can this type of equilibrium “non-credible” because the equilibrium is based on a strategy profile that the individual would like to change when they arrive at a future decision.
Game tree/ extensive form
Subgame
• A game comprising a portion of a longer game, starting from a non-initial node of the larger game
Backward induction
• Start at final subgames, to find Nash equilibrium for these subgames
• Move up the tree using these choices as predicted actions of later subgames.
Subgame perfect nash equilibrium SPNE
• An equilibrium found through backward induction
• A set of strategies that is Nash equilibrium at every single the games’ subgame
• Solution predicts stable outcomes because no player wants to deviate from her equilibrium strategies.
Airbus versus Boeing
• CEOs of Airbus and Boeing are in a strategic situation whether to introduce a new passenger jet