Intro to Game Theory
Revisiting the territory we have
covered
A look at the skeleton
• There is a fairly small set of ideas which we have
seen developed with a rich variety of examples.
• Today we look at this skeleton. The text develops
the examples and we have discussed a large
number of them.
• My advice for study. Read the assigned text and
readings carefully. Work problems. Try especially
to understand the assigned problems and the
worked out examples in the text.
What is a strategic game?
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Interacting players
Set of possible actions for each player
Action profiles
Preferences over action profiles
Example: The Stag Hunt
Player 2
Stag
Player 1
Hare
Stag
2 , 2
0, 1
Hare
1, 0
1 , 1
Actions Possible?
Action Profiles?
Preferences?
Dominating strategies.
• Action A strictly dominates action B for a
player if he prefers the outcome from doing A
to that from doing B, no matter what action
the other player takes.
• Does either strategy in the Stag Hunt
dominate the other strategy?
Another game
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 11
Strategy B
1 1, 0
1, 1
Does either strategy strictly dominate the other for
Player 1?
Does either strategy strictly dominate the other for
Player 2?
What are games like this called?
WeaklyDominating strategies.
• Action A weakly dominates action B for a
player if he likes the outcome from doing A to
at least as well as that from doing B, no
matter what action the other player takes.
• And for some actions by the other player, he
likes the outcome from A better.
How about this one?
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 10
Strategy B
1 0, 0
1, 1
Does either strategy weakly dominate the other
for Player 1?
Does either strategy strictly dominate the other for
Player 1?
Best response functions and
Nash Equilibrium
• The best response function for any player i, is
a function that maps the list of actions by
other players into the list of actions that are
best responses to what the others did.
– Sometimes there is only one best response.
• A Nash equilibrium is a set of actions by the
players such that each player’s action is a best
response to the actions of the other players.
Example: The Stag Hunt
Player 2
Stag
Player 1
Hare
Stag
2 , 2
0, 1
Hare
1, 0
1 , 1
B1(Stag)= {Stag}
B1(Hare)={Hare}
B2(Stag)={Stag}
B2(Hare)={Hare}
Nash equilibrium for 2-player game
• For a two-player game, a Nash equilibrium
consists of an action for each player such that
each player’s action is a best response to the
other player’s action.
• Method of stars works for games with finite
number of strategies.
• For game with continuum of strategies (e.g.
Cournot equilibrium, one calculates best
response function for each player, which typically
gives you two equations in two unknowns, which
you then solve.
Nash equlibrium for this game?
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 10
Strategy B
1 0, 0
1, 1
Nash equilibria for this game?
Player 2
Strategy A
Player 1
Strategy B
Strategy A
1 0 , 10
0, 11
Strategy B
1 1, 0
1, 1
What can you say about a Nash equilibrium in a
game where each player has a “strictly dominant
strategy”?
What do the best response functions look like if
there is a strictly dominant strategy?
Games with more than 2 players
• Each player’s best response can in general
depend on actions of all other players.
• A Nash equilibrium is a list of one action by
each player, such that each player’s action is a
best response to the actions of the other
players.
Mixed strategies
• Possible strategies include randomizing
between pure strategies.
• For this theory, we need to specify von
Neumann Morgenstern utilities.
• Players seek to maximize Bernoulli payoff
function which is the expected value of von
Neumann-Morgenstern utility.
• Here the intensity of preference as well as
order of preference matters.
Risk aversion, risk neutrality, risk loving
• With money prizes, von Neumann
Morgenstern preferences are given by a
function u(y) of prize amount y.
• Risk neutral if u’(y) is constant.
• Risk averse if u’( y) is decreasing.
• Risk loving if u’(y) is increasing.
Mixed Strategy Nash equilibrium
• A mixed strategy Nash equilibrium is a list of
mixed strategies for each player, such that
each player’s mixed strategy is a best response
to the other players’ mixed strategies. (Mixed
strategies are defined to include the pure
strategies as special cases.)
• Example: Matching pennies
Draw equilibrium diagram.
Change payoffs.
Extensive game with perfect
information?
• Set of players
• Set of terminal histories-a full history of a
game –possible courses of the game
• Player function: whose turn it is at each point
in the game
• Payoffs: to each player from each possible
terminal history
The entry game
Two players: Challenger and Incumbent.
Challenger moves first. Challenger either enters
the contest or stays out.
If challenger stays out, game ends. If challenger
gets payoff 1 and incumbent gets 2.
If Challenger enters, it is incumbent’s turn.
Either he yields or he fights. If he yields,
challenger gets 2, incumbent gets 1. If he fights,
Both get 0.
In the entry game
What are the possible strategies for the
entrant?
What are the possible strategies for the
incumbent?
What are the Nash equilibria?
What are the subgame perfect equilibria (um?)
What does this say about ``credible threats?’’
Coalitional Games
• Focus on what groups can accomplish if they
work together.
• Contrast to Nash equilibrium which focuses on
what individuals can do acting alone.
(sometimes known as non-cooperative game
theory)
Coalitional Game with transferable
payoffs
• A set of players N.
• A coalition S is a subset of N.
– Grand coalition is N itself.
• Coalitional game with transferable payoffs assigns
a value v(S) to every subset of S.
• An action for the coalition S is a distribution of
Its total value to its members.
Think of v(S) as an amount of “money” that the coalition
can earn on its own and can divide this money in any
way that adds to v(S).
The Core
• The core of a coalitional game is the set of
outcomes x (actions by the grand coalition)
such that no coalition has an action that all of
its members prefer to x.
A game with transferable payoffs and
no core: Majority redistribution game
• Players 1, 2, and 3.
• Non-empty subsets of N={1,2,3} are N,
{12}, {13}, {23},{1},{2},{3}.
• A cake whose total value is 1 is to be divided. Any
coalition that is a majority can choose how to divide it.
• Then v({12})=v({23})=v({13})=v({123})=1 and
v({1})=v({2})=v({3})=0.
• Actions available to any coalition are possible divisions
of the cake. For example, coalition{12} can choose
any division such that x1≥0, x2≥0, and x1+x2=1.
Coalitional Games without
transferable payoffs
• Roommate Assignment
• Marriage assignment
• College admissions
• What are the possible coalitions? What are
the payoffs?