Basic Game Theory
Economics 383 - Auction Theory
Instructor: Songzi Du
Simon Fraser University
September 10, 2015
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
1/7
Game Theory
Game theory studies the interaction of rational players:
make prediction about behavior
gives guidance on strategy
assess and improve the design (i.e., the rules) of the game
Basic ingredients of a game:
players
information
strategies
payoffs
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
2/7
An Example
You have a group presentation and an exam tomorrow.
Group presentation consists of you and a partner.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
3/7
An Example
You have a group presentation and an exam tomorrow.
Group presentation consists of you and a partner.
Expected grade for presentation,
both people prepare ⇒ 100
one person prepares ⇒ 92
nobody prepares ⇒ 84
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
3/7
An Example
You have a group presentation and an exam tomorrow.
Group presentation consists of you and a partner.
Expected grade for presentation,
both people prepare ⇒ 100
one person prepares ⇒ 92
nobody prepares ⇒ 84
Expected grade for exam,
study ⇒ 92
not study ⇒ 80
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
3/7
An Example
You have a group presentation and an exam tomorrow.
Group presentation consists of you and a partner.
Expected grade for presentation,
both people prepare ⇒ 100
one person prepares ⇒ 92
nobody prepares ⇒ 84
Expected grade for exam,
study ⇒ 92
not study ⇒ 80
Suppose you only have time either to prepare for the presentation or
to study for the exam (not both) tonight. Likewise for your partner.
What should you do?
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
3/7
Formal setup
Suppose there are n players, acting simultaneously.
Complete information
Payoff function:
Pi (S1 , S2 , . . . , Sn )
where Si ∈ Si is the strategy of player i.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
4/7
Formal setup
Suppose there are n players, acting simultaneously.
Complete information
Payoff function:
Pi (S1 , S2 , . . . , Sn )
where Si ∈ Si is the strategy of player i.
Strategy Si is player i’s best response to (S1 , . . . , Si−1 , Si+1 , . . . Sn ) if
Pi (S1 , . . . , Si−1 , Si , Si+1 , . . . , Sn ) ≥ Pi (S1 , . . . , Si−1 , Si0 , Si+1 , . . . , Sn )
for every strategy Si0 ∈ Si .
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
4/7
Dominant strategy and Nash equilibrium
A strategy profile is a combination of strategies for different players.
A strategy Si is the (weakly) dominant strategy for player i if
1
2
Si is a best response to every strategy profile of other players,
for every other strategy Si0 , there is a strategy profile of other players
such that Si is strictly better than Si0 for player i.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
5/7
Dominant strategy and Nash equilibrium
A strategy profile is a combination of strategies for different players.
A strategy Si is the (weakly) dominant strategy for player i if
1
2
Si is a best response to every strategy profile of other players,
for every other strategy Si0 , there is a strategy profile of other players
such that Si is strictly better than Si0 for player i.
A strategy Si is the strictly dominant strategy for player i if Si is
the strict best response to every strategy profile of other players.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
5/7
Dominant strategy and Nash equilibrium
A strategy profile is a combination of strategies for different players.
A strategy Si is the (weakly) dominant strategy for player i if
1
2
Si is a best response to every strategy profile of other players,
for every other strategy Si0 , there is a strategy profile of other players
such that Si is strictly better than Si0 for player i.
A strategy Si is the strictly dominant strategy for player i if Si is
the strict best response to every strategy profile of other players.
A strategy Si (strictly or weakly) dominates another strategy Si0 if Si
is (strictly or weakly) better than Si0 for player i given every strategy
profile of other players.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
5/7
Dominant strategy and Nash equilibrium
A strategy profile is a combination of strategies for different players.
A strategy Si is the (weakly) dominant strategy for player i if
1
2
Si is a best response to every strategy profile of other players,
for every other strategy Si0 , there is a strategy profile of other players
such that Si is strictly better than Si0 for player i.
A strategy Si is the strictly dominant strategy for player i if Si is
the strict best response to every strategy profile of other players.
A strategy Si (strictly or weakly) dominates another strategy Si0 if Si
is (strictly or weakly) better than Si0 for player i given every strategy
profile of other players.
A strategy profile (S1 , S2 , . . . , Sn ) is a Nash equilibrium if for every
player i, Si is a best response to the strategy profile
(S1 , . . . , Si−1 , Si+1 , . . . Sn ).
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
5/7
Examples
Battle of Sexes
Opera
Football
Opera
3, 2
0, 0
Football
0, 0
2, 3
Stag Hunt
Stag
Hare
Stag
5, 5
0, 3
Hare
3, 0
3, 3
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
6/7
A simple first-price auction
A single object for auction.
Two players, each has a value/willingness-to-pay vi for the object,
commonly known. Assume v1 = 2 and v2 = 1.
Each players submits a bid in {1, 2, 3}.
Highest bidder wins, pays his own bid, and gets the object.
If there is a tie, bidder 1 wins and pays.
ECON 383 (SFU)
Basic Game Theory
September 10, 2015
7/7
A simple first-price auction
A single object for auction.
Two players, each has a value/willingness-to-pay vi for the object,
commonly known. Assume v1 = 2 and v2 = 1.
Each players submits a bid in {1, 2, 3}.
Highest bidder wins, pays his own bid, and gets the object.
If there is a tie, bidder 1 wins and pays.
ECON 383 (SFU)
bid 1
bid 2
bid 3
bid 1
1, 0
0, -1
0, -2
bid 2
0, 0
0, 0
0, -2
bid 3
-1, 0
-1, 0
-1, 0
Basic Game Theory
September 10, 2015
7/7