EC941 - Game Theory Francesco Squintani Email:
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EC941 - Game Theory Francesco Squintani Email: f.squintaniwarwick.ac.uk 1 Lecture 1 Syllabus 1. Games in Strategic Form Definition and Solution Concepts Applications Readings: Chapter 2, 3, 12 2. Mixed Strategies Nash Equilibrium and
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01
EC941 - Game Theory Francesco Squintani
Email: f.squintani@warwick.ac.uk 1 Lecture 1<br>
Email: f.squintani@warwick.ac.uk 1 Lecture 1<br>
02
Syllabus 1. Games in Strategic Form
Definition and Solution Concepts
Applications
Readings: Chapter 2, 3, 12
2. Mixed Strategies
Nash Equilibrium and Rationalizability
Correlated Equilibrium
Readings: Chapter 4 2<br>
Definition and Solution Concepts
Applications
Readings: Chapter 2, 3, 12
2. Mixed Strategies
Nash Equilibrium and Rationalizability
Correlated Equilibrium
Readings: Chapter 4 2<br>
03
3. Bayesian Games
Definition
Information and Bayesian Games
Cournot Duopoly and Public Good Provision
Readings: Sections 9.1 to 9.6
4. Bayesian Game Applications
Juries and Information Aggregation
Auctions with Private Information
Readings: Sections 9.7 to 9.8 3<br>
Definition
Information and Bayesian Games
Cournot Duopoly and Public Good Provision
Readings: Sections 9.1 to 9.6
4. Bayesian Game Applications
Juries and Information Aggregation
Auctions with Private Information
Readings: Sections 9.7 to 9.8 3<br>
04
5. Extensive-Form Games
Definition
Subgame Perfection and Backward Induction
Applications
Readings: Chapters 5, 6 and 7
6. Extensive-Form Games with Imperfect Information
Definition
Spence Signalling Game
Crawford and Sobel Cheap Talk
Readings: Chapter 10 4<br>
Definition
Subgame Perfection and Backward Induction
Applications
Readings: Chapters 5, 6 and 7
6. Extensive-Form Games with Imperfect Information
Definition
Spence Signalling Game
Crawford and Sobel Cheap Talk
Readings: Chapter 10 4<br>
05
7. Repeated Games
Infinitely Repeated Games
Nash and Subgame-Perfect Equilibrium
Finitely Repeated Games
Readings: Chapter 14 and 15
8. Bargaining
Ultimatum Game and Hold Up Problem
Rubinstein Alternating Offer Bargaining
Nash Axiomatic Bargaining
Readings: Section 6.2 and Chapter 16 5<br>
Infinitely Repeated Games
Nash and Subgame-Perfect Equilibrium
Finitely Repeated Games
Readings: Chapter 14 and 15
8. Bargaining
Ultimatum Game and Hold Up Problem
Rubinstein Alternating Offer Bargaining
Nash Axiomatic Bargaining
Readings: Section 6.2 and Chapter 16 5<br>
06
9. Review Session
Solution of Past Exam Questions
Reference: An Introduction to Game Theory
Martin J. Osborne, Oxford University Press, 2003.
Assessment: Final Exam (100% of the grade)
Office Hours: Wednesday 9:00-11:00 – Room 1.123 6<br>
Solution of Past Exam Questions
Reference: An Introduction to Game Theory
Martin J. Osborne, Oxford University Press, 2003.
Assessment: Final Exam (100% of the grade)
Office Hours: Wednesday 9:00-11:00 – Room 1.123 6<br>
07
Structure of the Lecture Definition of Games in Strategic Form.
Solution Concepts
Nash Equilibrium, Dominance and Rationalizability.
Applications
Cournot Oligopoly, Bertrand Duopoly, Downsian
Electoral Competition, Vickrey Second Price Auction. 7<br>
Solution Concepts
Nash Equilibrium, Dominance and Rationalizability.
Applications
Cournot Oligopoly, Bertrand Duopoly, Downsian
Electoral Competition, Vickrey Second Price Auction. 7<br>
08
8 What is Game Theory? Game Theory is the formal study of strategic interactions.
A strategic interaction involves two or more agents. They maximize their payoffs and are aware that their opponents maximize payoffs.
Applications range from economics to politics, to biology and computer science.<br>
A strategic interaction involves two or more agents. They maximize their payoffs and are aware that their opponents maximize payoffs.
Applications range from economics to politics, to biology and computer science.<br>
09
Games in Strategic Form A game in strategic form is
a set of players: {1, 2, …, I}
for each player i, a set Si of strategies si
for each player i, preferences over the set of strategy profiles S={(s1 , …, sI )}, represented by u : S RI
(a strategy profile includes one strategy for each player). 9<br>
a set of players: {1, 2, …, I}
for each player i, a set Si of strategies si
for each player i, preferences over the set of strategy profiles S={(s1 , …, sI )}, represented by u : S RI
(a strategy profile includes one strategy for each player). 9<br>
10
Solution Concepts 10 A solution concept is a mathematical rule to find the solution of a game.
It allows the modeler to formulate a prediction on the play of the interaction she modeled as a game.
Today, we will study 3 solution concepts:
Nash Equilibrium
Dominance
Rationalizability<br>
It allows the modeler to formulate a prediction on the play of the interaction she modeled as a game.
Today, we will study 3 solution concepts:
Nash Equilibrium
Dominance
Rationalizability<br>
11
Nash Equilibrium A (pure-strategy) Nash equilibrium is a strategy profile
s∗ with the property that no player i can do better by
choosing a strategy different from si∗, given that every
other player j adheres to sj∗.
Definition The strategy profile s∗ is a Nash equilibrium
if, for every player i, ui(s∗) ≥ ui(s’i, s−i∗) for every strategy s’i
of player i. 11<br>
s∗ with the property that no player i can do better by
choosing a strategy different from si∗, given that every
other player j adheres to sj∗.
Definition The strategy profile s∗ is a Nash equilibrium
if, for every player i, ui(s∗) ≥ ui(s’i, s−i∗) for every strategy s’i
of player i. 11<br>
12
12 There are two main justifications for Nash Equilibrium:
Self-Enforcing Contract.
The players meet and agree before playing on the course of actions s∗. The contract s∗ is self-enforcing if no player has reasons to deviate if the others do not.
Learning Equilibrium Play.
The play s∗ is in equilibrium if no player i would deviate, were she to learn the opponents’ play s-i∗, because of communication, observation, or repetition.<br>
Self-Enforcing Contract.
The players meet and agree before playing on the course of actions s∗. The contract s∗ is self-enforcing if no player has reasons to deviate if the others do not.
Learning Equilibrium Play.
The play s∗ is in equilibrium if no player i would deviate, were she to learn the opponents’ play s-i∗, because of communication, observation, or repetition.<br>
13
Dominance A player’s strategy strictly dominates another one if it
gives a higher payoff, no matter of what other players do.
Definition Player i ’s strategy si strictly dominates strategy s’i
if ui (si , s−i) > ui (s’i , s−i) for every profile s−i of opponents’
strategies.
Theorem A strictly dominated strategy si is never part of any
Nash equilibrium s∗. 13<br>
gives a higher payoff, no matter of what other players do.
Definition Player i ’s strategy si strictly dominates strategy s’i
if ui (si , s−i) > ui (s’i , s−i) for every profile s−i of opponents’
strategies.
Theorem A strictly dominated strategy si is never part of any
Nash equilibrium s∗. 13<br>
14
A player’s strategy weakly dominates another strategy if it is always at least as good, and sometimes better.
Definition Player i ’s strategy si weakly dominates her strategy s’i if
ui (si , s−i) ≥ ui(s’i , s−i) for every profile s−i of opponents’ strategies
ui(si , s−i) > ui(s’i , s−i) for some profile s−i of opponents’ strategies.
Note There exist games with Nash Equilibria s∗ that include weakly dominated strategies si for some player i. 14<br>
Definition Player i ’s strategy si weakly dominates her strategy s’i if
ui (si , s−i) ≥ ui(s’i , s−i) for every profile s−i of opponents’ strategies
ui(si , s−i) > ui(s’i , s−i) for some profile s−i of opponents’ strategies.
Note There exist games with Nash Equilibria s∗ that include weakly dominated strategies si for some player i. 14<br>
15
There are two Nash Equilibria: (A,C) and (B,D).
The Nash Equilibrium (B,D) is weakly dominated. 15 Player 2
C D Player 1
A
B 1, 1 0, 0 0, 0 0, 0<br>
The Nash Equilibrium (B,D) is weakly dominated. 15 Player 2
C D Player 1
A
B 1, 1 0, 0 0, 0 0, 0<br>
16
Rationalizability is defined via iterated deletion of strictly
dominated strategies.
Consider a finite game G = (I, S, u).
For each player i, and round t = 1, . . . , T, iteratively
define the set Xit of strategies of player i as follows.
Xi1 = Si (start with the set of all possible strategies). Rationalizability 16<br>
dominated strategies.
Consider a finite game G = (I, S, u).
For each player i, and round t = 1, . . . , T, iteratively
define the set Xit of strategies of player i as follows.
Xi1 = Si (start with the set of all possible strategies). Rationalizability 16<br>
17
For each t = 0, . . . , T − 1, Xit+1 is a subset of Xit such that every strategy of player i in Xit that is not in Xit+1 is strictly dominated in the game where the set of strategy of each player j is reduced to Xjt
(in each round, delete all strictly dominated strategies).
The final index T is such that no strategy in XiT is strictly dominated in the game where the set of strategies of each player j is reduced to XjT
(proceed until no strategy is strictly dominated).
The set XiT is the set of rationalizable strategies of player i. 17<br>
(in each round, delete all strictly dominated strategies).
The final index T is such that no strategy in XiT is strictly dominated in the game where the set of strategies of each player j is reduced to XjT
(proceed until no strategy is strictly dominated).
The set XiT is the set of rationalizable strategies of player i. 17<br>
18
18 Rationalizability is justified by common knowledge of rationality.
Each player is rational: She does not play strictly dominated strategies.
Each player knows that every player is rational: She can reduce the game by deleting all players’ strictly dominated strategies from her model of the interaction (the game).
Each player knows that every player knows that every player is rational: She deletes all strictly dominated strategies in the reduced game.
The procedure is iterated until it stops.<br>
Each player is rational: She does not play strictly dominated strategies.
Each player knows that every player is rational: She can reduce the game by deleting all players’ strictly dominated strategies from her model of the interaction (the game).
Each player knows that every player knows that every player is rational: She deletes all strictly dominated strategies in the reduced game.
The procedure is iterated until it stops.<br>
19
Best Response Correspondences The best response correspondence Bi of player i assigns
to each profile s-i of opponents’ strategies, the set of
player i ’s strategies that maximizes her payoff.
Definition The best response correspondence Bi of player i is:
Bi (s−i) = {si in Si : ui(si, s−i) ≥ ui(s’i, s−i) for all s’i in Si}.
Proposition The strategy profile s∗ is a Nash equilibrium of a
game G=(I, S, u) if and only if every player’s strategy is a best
response to the other players’ strategies: s∗i belongs to Bi(s∗−i ) for
every player i. 19<br>
to each profile s-i of opponents’ strategies, the set of
player i ’s strategies that maximizes her payoff.
Definition The best response correspondence Bi of player i is:
Bi (s−i) = {si in Si : ui(si, s−i) ≥ ui(s’i, s−i) for all s’i in Si}.
Proposition The strategy profile s∗ is a Nash equilibrium of a
game G=(I, S, u) if and only if every player’s strategy is a best
response to the other players’ strategies: s∗i belongs to Bi(s∗−i ) for
every player i. 19<br>
20
Prisoner’s Dilemma Two prisoners are separately interviewed. By accusing the
other suspect, one’s prison term is reduced. But if they
both stayed quiet, they would not be incarcerated.
Players: The two suspects.
Strategies: Each player’s set of strategy is {Quiet, Fink}.
Preferences: Suspect 1’s ordering of the strategy profiles, from best to worst is (F, Q), (Q, Q), (F, F), (Q, F).
Suspect 2’s ordering is (Q, F), (Q, Q), (F, F), (F, Q). 20<br>
other suspect, one’s prison term is reduced. But if they
both stayed quiet, they would not be incarcerated.
Players: The two suspects.
Strategies: Each player’s set of strategy is {Quiet, Fink}.
Preferences: Suspect 1’s ordering of the strategy profiles, from best to worst is (F, Q), (Q, Q), (F, F), (Q, F).
Suspect 2’s ordering is (Q, F), (Q, Q), (F, F), (F, Q). 20<br>
21
Suspect 2
Quiet Fink Suspect 1
Quiet
Fink 2, 2 0, 3 3, 0 1, 1 21<br>
Quiet Fink Suspect 1
Quiet
Fink 2, 2 0, 3 3, 0 1, 1 21<br>
22
Solutions of Prisoner’s Dilemma
Quiet Fink Quiet
Fink 2, 2 0, 3 3, 0 1, 1 Fink is the best response of each player, regardless of what
the other player does. Fink is the strictly dominant and
rationalizable strategy. (Fink, Fink) is the Nash Equilibrium. 22<br>
Quiet Fink Quiet
Fink 2, 2 0, 3 3, 0 1, 1 Fink is the best response of each player, regardless of what
the other player does. Fink is the strictly dominant and
rationalizable strategy. (Fink, Fink) is the Nash Equilibrium. 22<br>
23
Bach or Stravinsky Two daters would rather be together than separate, but
dater 1 prefers Bach and dater 2 prefers Stravinsky.
Players: The two daters.
Strategies: Each dater’s strategy set is {Bach, Stravinsky}.
Preferences: Dater 1’s ordering of the strategy profiles, from best to worst is (B, B), (S, S), (B, S) = (S, B).
Dater 2’s ordering is (S, S), (B, B), (S, B) = (B, S). 23<br>
dater 1 prefers Bach and dater 2 prefers Stravinsky.
Players: The two daters.
Strategies: Each dater’s strategy set is {Bach, Stravinsky}.
Preferences: Dater 1’s ordering of the strategy profiles, from best to worst is (B, B), (S, S), (B, S) = (S, B).
Dater 2’s ordering is (S, S), (B, B), (S, B) = (B, S). 23<br>
24
Dater 2
Bach Stravinsky Dater 1
Bach
Stravinsky 2, 1 0, 0 0, 0 1, 2 If they can coordinate, either the two daters go to Bach’s
concert or to Stravinsky’s concert. 24<br>
Bach Stravinsky Dater 1
Bach
Stravinsky 2, 1 0, 0 0, 0 1, 2 If they can coordinate, either the two daters go to Bach’s
concert or to Stravinsky’s concert. 24<br>
25
Solutions of Bach or Stravinsky Bach
Stravinsky 2, 1 0, 0 0, 0 1, 2 For each player, B is the best response to B, and S is the best response to S. There are two Nash Equilibria, (B, B) and (S, S). All strategies are rationalizable, and none is dominated. 25 Bach Stravinsky<br>
Stravinsky 2, 1 0, 0 0, 0 1, 2 For each player, B is the best response to B, and S is the best response to S. There are two Nash Equilibria, (B, B) and (S, S). All strategies are rationalizable, and none is dominated. 25 Bach Stravinsky<br>
26
Matching Pennies Player 1 wins if the coins are matched.
Player 2 wins if they are not matched.
Players: The two players.
Strategies: Each player’s set of actions is {Head, Tail}.
Preferences: Player 1’s ordering of the strategy profiles, from best to worst, is (H, T) = (T, H), (H, H) = (T, T).
Player 2’s ordering is (H, H) = (T, T), (H, T) = (T, H). 26<br>
Player 2 wins if they are not matched.
Players: The two players.
Strategies: Each player’s set of actions is {Head, Tail}.
Preferences: Player 1’s ordering of the strategy profiles, from best to worst, is (H, T) = (T, H), (H, H) = (T, T).
Player 2’s ordering is (H, H) = (T, T), (H, T) = (T, H). 26<br>
27
Player 2
Head Tail Player 1
Head
Tail -1, 1 1, -1 There is no sure way to win for either of the players. -1, 1 1, -1 27<br>
Head Tail Player 1
Head
Tail -1, 1 1, -1 There is no sure way to win for either of the players. -1, 1 1, -1 27<br>
28
Solutions of Matching Pennies For player 1, H is the best response to T and viceversa, for player 2, H is the best response to H and T is the best
response to T. All strategies are rationalizable and none is dominated. There are no Nash Equilibria. 28 Head Tail Head
Tail -1, 1 1, -1 -1, 1 1, -1<br>
response to T. All strategies are rationalizable and none is dominated. There are no Nash Equilibria. 28 Head Tail Head
Tail -1, 1 1, -1 -1, 1 1, -1<br>
29
Cournot Oligopoly A good is produced by n firms.
Firm i’s cost of producing qi units is Ci(qi).
Ci is an increasing function.
The firms' total output is Q = q1 + … + qn.
The market price is P(Q).
P is the inverse demand function, decreasing if positive. 29<br>
Firm i’s cost of producing qi units is Ci(qi).
Ci is an increasing function.
The firms' total output is Q = q1 + … + qn.
The market price is P(Q).
P is the inverse demand function, decreasing if positive. 29<br>
30
Firm i’s revenue is qi P(q1 + … + qn).
Firm i’s profit is revenue minus cost:
pi(q1 + … + qn) = qi P(q1 + … + qn) - Ci (qi).
Ci (qi) = cqi, i=1, …, n.
P (Q) = a - Q if a > Q, P(Q) = 0 if a < Q.
pi(q1, …, qn) = qi [a – (q1 + … + qn)] - cqi. Linear Costs and Demand 30<br>
Firm i’s profit is revenue minus cost:
pi(q1 + … + qn) = qi P(q1 + … + qn) - Ci (qi).
Ci (qi) = cqi, i=1, …, n.
P (Q) = a - Q if a > Q, P(Q) = 0 if a < Q.
pi(q1, …, qn) = qi [a – (q1 + … + qn)] - cqi. Linear Costs and Demand 30<br>
31
To find the Best Response Functions, differentiate pi with respect to qi, set it equal to zero, and obtain:
dpi (q1, …, qn)/dqi = a – qi – (q1 + … + qn) – c = 0.
Best Response functions:
bi (qi) = [a – (q1 + … + qi-1 + qi+1 +…+ q n) – c]/2.
To find the Nash equilibria, we solve the system of best-response functions.
Because this system is linear and symmetric, we equalize qi* across i = 1,…,n:
qi* = bi (qi*) = [a – (n-1) qi* – c]/2. 31<br>
dpi (q1, …, qn)/dqi = a – qi – (q1 + … + qn) – c = 0.
Best Response functions:
bi (qi) = [a – (q1 + … + qi-1 + qi+1 +…+ q n) – c]/2.
To find the Nash equilibria, we solve the system of best-response functions.
Because this system is linear and symmetric, we equalize qi* across i = 1,…,n:
qi* = bi (qi*) = [a – (n-1) qi* – c]/2. 31<br>
32
32 Solving the above equation, we find that the Nash equilibrium quantity is:
qi* = [a – c]/(n+1).
Substituting in the formula for the price, we find that the Nash equilibrium price is:
pi (qi*) = a – Q* = a – n[a – c]/(n+1)
= [a + nc]/(n+1).
The Nash equilibrium profits are:
pi (qi*) = qi*[a – Q*] – cqi*
= [a – c ]2/(n-1)2.<br>
qi* = [a – c]/(n+1).
Substituting in the formula for the price, we find that the Nash equilibrium price is:
pi (qi*) = a – Q* = a – n[a – c]/(n+1)
= [a + nc]/(n+1).
The Nash equilibrium profits are:
pi (qi*) = qi*[a – Q*] – cqi*
= [a – c ]2/(n-1)2.<br>
33
[a – c]/3 [a – c]/3 q1 q2 b1(q2) b2(q1) (q1*, q2*) qi* = [a – c]/3, i = 1,2. With n = 2,
b1(q2) = [a – q2– c]/2.
b2(q1) = [a – q1– c]/2. 33<br>
b1(q2) = [a – q2– c]/2.
b2(q1) = [a – q1– c]/2. 33<br>
34
Bertrand Competition Unlike Cournot competition, firms compete in prices.
The demand function is denoted by D,
if the good is available at the price p, then the total amount demanded is D(p).
The firm setting the lowest price sells to all the market. 34<br>
The demand function is denoted by D,
if the good is available at the price p, then the total amount demanded is D(p).
The firm setting the lowest price sells to all the market. 34<br>
35
Linear Costs and Demand Ci(qi) = cqi, i=1, …, n.
D(p) = a – p if a > p, D(p) = 0 if a < p.
Let pi = min {pj, j different from i}.
The profit is:
pi(p1, …, pn) = (pi – c)(a - pi) if pi < pi,
pi(p) = (pi – c)(a - pi)/|{k : pk = pi}| if pi = pi,
pi(p) = 0 if pi > pi. 35<br>
D(p) = a – p if a > p, D(p) = 0 if a < p.
Let pi = min {pj, j different from i}.
The profit is:
pi(p1, …, pn) = (pi – c)(a - pi) if pi < pi,
pi(p) = (pi – c)(a - pi)/|{k : pk = pi}| if pi = pi,
pi(p) = 0 if pi > pi. 35<br>
36
Best-Response Correspondence Suppose that there only two firms, so that pi= pj.
If pj < c, then pi(p) < 0 for pi < pj,
pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}. pi pi pj c pm 36<br>
If pj < c, then pi(p) < 0 for pi < pj,
pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}. pi pi pj c pm 36<br>
37
If pj = c, then pi(p) < 0 for pi < pj,
pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}.
If pj > pm, then bi(p) = {pm}. pi pj c pm pi pi pj c pm pi 37<br>
pi(p) = 0 for pi > pj : bi(p) = {pi : pi > pj}.
If pj > pm, then bi(p) = {pm}. pi pj c pm pi pi pj c pm pi 37<br>
38
If c < pj < pm then pi(p) increases in pj, but discontinuously drops at pi = pj. So, bi(p) = f.
The best response correspondence is empty. pi pi pj c 38<br>
The best response correspondence is empty. pi pi pj c 38<br>
39
In sum, the best-response correspondence is:
bi(p) = {pi : pi > pj}, if pj < c,
bi(p) = {pi : pi > pj}, if pj = c,
bi(p) = f, if c < pj < pm,
bi(p) = {pm}, if pj > pm.
The Nash equilibrium is pi = c, for all i = 1,…,n.
Intuitively, selling at any price pi < c yields negative profit.
If the lowest industry price were pj > c, then firm i sells to
the whole industry at any price pi with c < pi < pj.
In equilibrium, pi = c, for all i. 39<br>
bi(p) = {pi : pi > pj}, if pj < c,
bi(p) = {pi : pi > pj}, if pj = c,
bi(p) = f, if c < pj < pm,
bi(p) = {pm}, if pj > pm.
The Nash equilibrium is pi = c, for all i = 1,…,n.
Intuitively, selling at any price pi < c yields negative profit.
If the lowest industry price were pj > c, then firm i sells to
the whole industry at any price pi with c < pi < pj.
In equilibrium, pi = c, for all i. 39<br>
40
Downsian Electoral Competition The players are 2 candidates in an election.
A strategy is a real number x, representing a policy on the left-right political ideology spectrum.
After the candidates choose policies, each citizen votes for the candidate with the policy she prefers.
The candidate who obtains the most votes wins. Candidates care only about winning. 40<br>
A strategy is a real number x, representing a policy on the left-right political ideology spectrum.
After the candidates choose policies, each citizen votes for the candidate with the policy she prefers.
The candidate who obtains the most votes wins. Candidates care only about winning. 40<br>
41
41 The voters are a continuum with diverse ideologies, with cumulative distribution F.
For any k, a voter with ideology y is indifferent between the policies y - k and y + k.
The median m is such that 1/2 of voters’ has ideologies y > m, and 1/2 has ideologies y < m. So, F (m) = 1/2.<br>
For any k, a voter with ideology y is indifferent between the policies y - k and y + k.
The median m is such that 1/2 of voters’ has ideologies y > m, and 1/2 has ideologies y < m. So, F (m) = 1/2.<br>
42
Fix the policy x2 of candidate 2 and consider 1’s choice.
Suppose that x2 < m, the case for x2 > m is symmetric.
If candidate 1 chooses x1 < x2 then she wins the votes of citizens with ideology y < ½ ( x1 + x2 ).
Because ½ ( x1 + x2 ) < x2 < m, it follows that
F(½ ( x1 + x2 ) ) < ½, so that candidate 1 wins less than
½ of the votes, and loses the election. Best Response Functions 42<br>
Suppose that x2 < m, the case for x2 > m is symmetric.
If candidate 1 chooses x1 < x2 then she wins the votes of citizens with ideology y < ½ ( x1 + x2 ).
Because ½ ( x1 + x2 ) < x2 < m, it follows that
F(½ ( x1 + x2 ) ) < ½, so that candidate 1 wins less than
½ of the votes, and loses the election. Best Response Functions 42<br>
43
If x1 > x2, then candidate 1 wins the votes of citizens with ideology y > ½ ( x1 + x2 ).
She wins more or less than ½ of the votes if and only if
1 – F(½ ( x1 + x2 )) > ½.
In this case, she wins the election.
This is equivalent to ½ ( x1 + x2 ) < m, i.e. x1 < 2m - x2.
So, b1 (x2) = {x1 : x2 < x1 < 2m - x2 } for x2 < m. 43<br>
She wins more or less than ½ of the votes if and only if
1 – F(½ ( x1 + x2 )) > ½.
In this case, she wins the election.
This is equivalent to ½ ( x1 + x2 ) < m, i.e. x1 < 2m - x2.
So, b1 (x2) = {x1 : x2 < x1 < 2m - x2 } for x2 < m. 43<br>
44
For x2 > m, b1 (x2) = {x1 : 2m - x2 < x1 < x2}.
If x2 = m, then player 1 loses the election unless she plays x1 = m. So b1 (m) = {m}.
By using the best response correspondences the unique Nash Equilibrium is (m, m). The candidates’ political platforms converge to the median policy. 44<br>
If x2 = m, then player 1 loses the election unless she plays x1 = m. So b1 (m) = {m}.
By using the best response correspondences the unique Nash Equilibrium is (m, m). The candidates’ political platforms converge to the median policy. 44<br>
45
Intuitively, consider any pair of platforms (x1, x2) other than (m, m). One candidate can win the election by deviating and locating e closer to m than x2.
Hence (x1, x2) is not a Nash Equilibrium.
If instead x1=m, then candidate 2 loses the election for sure unless she plays x2 = m.
Hence (m, m) is a Nash Equilibrium. 45<br>
Hence (x1, x2) is not a Nash Equilibrium.
If instead x1=m, then candidate 2 loses the election for sure unless she plays x2 = m.
Hence (m, m) is a Nash Equilibrium. 45<br>
46
Vickrey Second-Price Auctions In an “English” auction, n bidders submit increasing bids for a good, until only one is left, who wins the auction.
The price paid by the last bidder is her last bid.
Suppose each bidder’s valuation of the good is independent of the other bidders’ values.
For example, Vickrey’s model applies when the good is a work of art, but not when it is a oil field. 46<br>
The price paid by the last bidder is her last bid.
Suppose each bidder’s valuation of the good is independent of the other bidders’ values.
For example, Vickrey’s model applies when the good is a work of art, but not when it is a oil field. 46<br>
47
The English auction is equivalent to a sealed-bid auction, in which each bidder decides, before bidding begins, the most she is willing to bid.
To win, the bidder with the highest valuation needs to bid slightly more than the second highest maximal bid.
If the bidding increment is small, the price the winner pays equals the second highest maximal bid. 47<br>
To win, the bidder with the highest valuation needs to bid slightly more than the second highest maximal bid.
If the bidding increment is small, the price the winner pays equals the second highest maximal bid. 47<br>
48
Second-Price Auction Game Players: n bidders. Bidder i’s valuation is vi, we order v1> … > vn > 0, without loss of generality.
Strategies: bidder i’s maximal bid is bi.
Let bi = max {bj : j different from i}.
Payoffs: ui(b1, … ,bn) = vi - bi if bi > bi
0 if bi < bi 48<br>
Strategies: bidder i’s maximal bid is bi.
Let bi = max {bj : j different from i}.
Payoffs: ui(b1, … ,bn) = vi - bi if bi > bi
0 if bi < bi 48<br>
49
1. (b*1,… , b*n) = (v1, …, vn).
Bidder 1 wins the object, payoff: v1 – b*2 = v1 – v2 > 0.
If bidding b1 < v2, she loses the object, the payoff is 0.
If bidding b1 > v2, her payoff is v1 – v2 > 0.
The payoff of bidders i = 2, …, n is 0.
If bidding bi > v1, the payoff is vi – b1 = vi – v1 < 0.
If bidding bi < v1, she loses the object, the payoff is 0. Nash Equilibria 49<br>
Bidder 1 wins the object, payoff: v1 – b*2 = v1 – v2 > 0.
If bidding b1 < v2, she loses the object, the payoff is 0.
If bidding b1 > v2, her payoff is v1 – v2 > 0.
The payoff of bidders i = 2, …, n is 0.
If bidding bi > v1, the payoff is vi – b1 = vi – v1 < 0.
If bidding bi < v1, she loses the object, the payoff is 0. Nash Equilibria 49<br>
50
2. (b*1,… , b*n) = (v1, 0,… , 0)
Bidder 1 wins the object, her payoff is v1.
The payoff of bidders i = 2, …, n is 0.
If bidding bi > v1, the payoff is vi – b1 = vi – v1 < 0.
If bidding bi < v1, she loses the object, the payoff is 0.
3. (b*1,… , b*n) = (v2, v1 , 0,… , 0)
Bidder 2 wins the object, payoff v2 – b1 = v2 – v2 = 0.
To win, bidder i = 1, 3, …, n must bid bi > b2 = v1,
so the payoff is vi – b1 < vi – v1 < 0.
Any of these bidders’ payoff is at least as good if losing the good. 50<br>
Bidder 1 wins the object, her payoff is v1.
The payoff of bidders i = 2, …, n is 0.
If bidding bi > v1, the payoff is vi – b1 = vi – v1 < 0.
If bidding bi < v1, she loses the object, the payoff is 0.
3. (b*1,… , b*n) = (v2, v1 , 0,… , 0)
Bidder 2 wins the object, payoff v2 – b1 = v2 – v2 = 0.
To win, bidder i = 1, 3, …, n must bid bi > b2 = v1,
so the payoff is vi – b1 < vi – v1 < 0.
Any of these bidders’ payoff is at least as good if losing the good. 50<br>
51
Weakly Dominant Solution The Nash Equilibrium (b*1,… , b*n) = (v1, …, vn) is the
unique weakly dominant solution. bi < vi bi < bi or
bi = bi & i wins bi < bi < vi or
bi=bi & i loses bi > vi vi - bi vi - bi vi - bi 0 0 0 bi = vi 51<br>
unique weakly dominant solution. bi < vi bi < bi or
bi = bi & i wins bi < bi < vi or
bi=bi & i loses bi > vi vi - bi vi - bi vi - bi 0 0 0 bi = vi 51<br>
52
bi > vi bi < vi vi < bi < bi or
bi =bi & i wins vi - bi vi - bi vi – bi (< 0) 0 0 0 bi > bi or
bi =bi & i loses In sum, bidding bi = vi yields at least as high a payoff as bidding bi > vi or bi < vi for any opponents’ bids. bi = vi 52<br>
bi =bi & i wins vi - bi vi - bi vi – bi (< 0) 0 0 0 bi > bi or
bi =bi & i loses In sum, bidding bi = vi yields at least as high a payoff as bidding bi > vi or bi < vi for any opponents’ bids. bi = vi 52<br>
53
Definition of Games in Strategic Form.
Definition of Solution Concepts
Nash Equilibrium, Dominance and Rationalizability.
Applications
Cournot Oligopoly, Bertrand Duopoly, Downsian
Electoral Competition, Vickrey Second Price Auction. Summary of the Lecture 53<br>
Definition of Solution Concepts
Nash Equilibrium, Dominance and Rationalizability.
Applications
Cournot Oligopoly, Bertrand Duopoly, Downsian
Electoral Competition, Vickrey Second Price Auction. Summary of the Lecture 53<br>
54
Preview Next Lecture Mixed Strategies.
Nash Equilibrium and Rationalizability.
Correlated Equilibrium. 54<br>
Nash Equilibrium and Rationalizability.
Correlated Equilibrium. 54<br>