1. UNIT II: The Basic Theory
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UNIT II: The Basic Theory
Zero-sum Games
Nonzero-sum Games
Nash Equilibrium: Properties and Problems
Bargaining Games
Bargaining and Negotiation
Review
Midterm 7/17
7/8

2. Nonzero-sum Games Examples: Bargaining Duopoly International Trade
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Nonzero-sum Games
Examples:
Bargaining
Duopoly
International Trade

3. Nonzero-sum Games The Essentials of a Game
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Nonzero-sum Games
The Essentials of a Game
Eliminating Dominated Strategies
Best Response
Nash Equilibrium
Duopoly: An Application
Solving the Game
Existence of Nash Equilibrium
Properties and Problems
See: Gibbons, Game Theory for Applied Economists (1992): 1-51.

4. The Essentials of a Game
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The Essentials of a Game
1. Players: We require at least 2 players (Players choose actions and receive payoffs.)
2. Actions: Player i chooses from a finite set of actions, S = {s1,s2,…..,sn}. Player j chooses from a finite set of actions T = {t1,t2,……,tm}.
3A. Payoffs: We define Pi(s,t) as the payoff to player i, if Player i chooses s and player j chooses t. We allow that Pi(s,t) + Pj(s,t) = 0.
4. Information: What players know (believe) when choosing actions.
NONZERO-SUM

5. Eliminating Dominated Strategies
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Eliminating Dominated Strategies
L M R
R is strictly dominated by M, so the game can be reduced to
Now, B is strictly dominated by T ... T B
1, ,2 0,1
0, ,1 2,0 T B
1, ,2
0, ,1 T
1, ,2
(T, M)

6. Nash Equilibrium S1 S1 S2 S2 S3 S3 (S3,T3) T1 T2 T3 0,4 4,0 5,3
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Nash Equilibrium
Definitions
Best Response Strategy: a strategy, s*, is a best response strategy, iff the payoff to (s*,t) is at least as great as the payoff to (s,t) for all s.
T1 T2 T3 S1 S2 S3
0,4 4,0 5,3
4,0 0,4 5,3
3,5 3,5 6,6
Nash Equilibrium: a set of best response strategies (one for
each player), (s*, t*)
such that s* is a best
response to t* and
t* is a b.r. to s*.
(S3,T3) S1 S2 S3
Now let me pause and see if there are any questions.

7. Nash Equilibrium S1 S1 S2 S2 S3 S3 T1 T2 T3
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Nash Equilibrium
T1 T2 T3
Nash equilibrium need not be
efficient. S1 S2 S3
4,4 2,3 1,5
3,2 1,1 0,0
5,1 0,0 3,3 S1 S2 S3
This is similar to the PD, but w/o dominant strategies.

8. Nash Equilibrium S1 S1 S2 S2 S3 S3 T1 T2 T3
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Nash Equilibrium
T1 T2 T3
Nash equilibrium need not be
unique.
A COORDINATION PROBLEM
What is the effect of repeated play? S1 S2 S3
1,1 0,0 0,0
0,0 1,1 0,0
0,0 0,0 1,1 S1 S2 S3
GC PS TS
A relatively simple problem to solve, if the players can communicate.

9. Duopoly
Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost).
Market demand can be described by:
P(rice) = 8 – Q(uantity)
Where Q is total industry output (Q = q1 + q2)

10. Duopoly
Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost).
For each firm, Profit (p) = Total Revenue – Total Cost
= Pq – q
Each firm will choose a level of output q, to maximize its profit, taking into account what it expects the other firm to produce.

11. Duopoly P = 8 - Q Demand Condition FOC: 6 - 2q1 - q2 = 0
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Duopoly
Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output.
P = 8 - Q Demand Condition
p1 = Total Revenue – Total Costq1 - 2q1
= 6q1 - q12 - q2q1
FOC: q1 - q2 = 0 q
q1* = 3 – 1/2q2 q1 3
q1* Firm 1’s profit maximizing output as a function of Firm 2’s output.
6 q2

12. 5/14/2019
Duopoly
Each firm’s profit maximizing level of output, q*, is a function of the other firm’s output.
P = 8 - (q1+q2)
p1 = Pq1 - 2q1 = [8 - (q1+q2)]q1 - 2q1
= 6q1 - q12 - q2q1
= 6 - 2q1 - q2 = 0 q
q1* = 3 – 1/2q2 q1 3
dp1
q1* Firm 1’s profit maximizing output as a function of Firm 2’s output.
FOC:
dq1
6 q2

13. 5/14/2019
Duopoly
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*.
q2* = 3 - 1/2q1
q1* = 3 - 1/2q2 q1
q1* = 2
A Nash Equilibrium exists in an oligopolistic market, if each firm is basing its pmax output on a correct assumption (consistent) about the rivals’ behavior.
On Dynamics: We arrive at the NE in logical time not historical time. Convergence to the NE is not the result of a series of actions but a series of conjectures, or beliefs. I think-that you think-that I think …
[I]f game theory is to provide a unique solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).
q2* = 2 q2

14. 5/14/2019
Duopoly
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*. q1
q1*
Is this the best they can do?
If Firm 1 reduces its output while Firm 2 continues to produce q2*, the price rises and Firm 2’s profits increase.
q2* q2

15. 5/14/2019
Duopoly
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*. q1
q1*
Is this the best they can do?
If Firm 2 reduces its output while Firm 1 continues to produce q1*, the price rises and Firm 1’s profits increase.
q2* q2

16. 5/14/2019
Duopoly
A Nash Equilibrium is a pair of “best responses,” such that q1* is a best response to q2* and q2* is a best response to q1*. q1
q1*
Is this the best they can do?
There are a range of outcomes to the SW that make both firms better off.
The duopolists have an incentive to collude: to restrict their output – below the NE level – and increase their profits.
What is socially optimal
q2* q2

17. Duopoly
Consider a market in which two identical firms can produce a good with marginal cost = $2 per unit (assume no fixed cost).
Market demand can be described by: P(rice) = 8 – Q(uantity)
Where Q is total industry output (Q = q1 + q2)
Assume that each firm can only choose a discrete quantity 0, 1, 2, 3, 4. What is the matrix form of this game?
What is the best response for Firm 1, if it thinks Firm 2 will produce 3 units of output? If it thinks Firm 2 will produce 4 units of output?
Find the Nash equilibrium in the market.

18. Duopoly P = 8 - Q Q = q1 + q2 pi = (P - 2)qi q1 q2 P p1 p2 0 0 8 0 0

19. Duopoly
q1 q2 P p1 p2 1 2 3 4
0,0 0, , ,9 0,8
5,0 4, , ,6 1,4
8,0 6, , ,3 0,0
9,0 6, , ,0 -3,-4
8,0 4,1 0,0 -4,-3 -8,-8

20. Duopoly 4 3 Best Response Functions 2 1 NE 0 1 2 3 4 q1 q2
q1 q2 P p1 p2 4 3 2 1
q2 (q1)
Best Response Functions NE
q1 (q2) q1

21. 5/14/2019
Duopoly
In a Nash Equilibrium, no firm can increase its profits by changing its output unilaterally. It is strategically stable or self-enforcing agreement.
A Nash Equilibrium does not necessarily maximize joint (or industry) profits. But improving the situation creates an enforcement problem.
In general, we observe a tension between
Collusion: maximize joint profits
Competition: capture a larger share of the pie
Examples of Collusion

22. Solving the 2x2 Game T1 T2 3. Prisoner’s Dilemma 4. Stag Hunt S1
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Solving the 2x2 Game
T T2
3. Prisoner’s Dilemma
4. Stag Hunt
5. Chicken
6. Battle of the Sexes S1 S2
x1,x w1,w2
z1,z y1,y2

23. Solving the Game T1 T2 3. Prisoner’s Dilemma S1 3,3 0,5 NE = {(S2,T2)}
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Solving the Game
T T2
3. Prisoner’s Dilemma
NE = {(S2,T2)} S1 S2
3, ,5
5, ,1

24. Solving the Game T1 T2 4. Stag Hunt (also, Assurance Game) S1 5,5 0,3
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Solving the Game
T T2
4. Stag Hunt
(also, Assurance Game)
NE = {(S1,T1), (S2,T2)} S1 S2
5, ,3
3, ,1

25. Solving the Game T1 T2 5. Chicken (also Hawk/Dove) S1 3,3 1,5
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Solving the Game
T T2
5. Chicken
(also Hawk/Dove)
NE = {(S1,T2), (S2,T1)} S1 S2
3, ,5
5, ,0

26. Solving the Game O F 6. Battle of the Sexes O 5,3 0,0
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Solving the Game
O F
6. Battle of the Sexes
NE = {(O,O), (F,F)} O F
5, ,0
0, ,5
Find the mixed strategy Nash Equilibrium

27. Solving the Game O F O 5,3 0,0 F 0,0 3,5 Equalizers
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Solving the Game
O F
Let (p,1-p) = prob1(O, F )
(q,1-q) = prob2(O, F )
Then
EP1(O) = 5q
EP1(F) = 3-3q
q* = 3/8
EP2(O) = 3p
EP2(F) = 5-5p
p* = 5/8 O F
5, ,0
0, ,5
Equalizers
NE = {(1, 1); (0, 0); (5/8, 3/8)} ); (0, 0); (5/8, 3/8 Game 6.

28. Solving the Game q*(p) Game 6. q OPERA 1 3/8 FIGHT 0 0 5/8 1 p
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Solving the Game q
OPERA
3/8
FIGHT
q*(p)
if p<5/8, then Player 2’s best response is q* = 0 (FIGHT)
if p>5/8 q* = 1 (OPERA)
/ p
Game 6.

29. Solving the Game q*(p) p*(q) NE = {(1, 1); (0, 0); (5/8, 3/8)} Game 6.
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Solving the Game q
OPERA
3/8
FIGHT
q*(p)
p*(q)
/ p
NE = {(1, 1); (0, 0); (5/8, 3/8)}
Game 6.
(p, q); (p, q)

30. Solving the Game The Battle of the Sexes FIGHT OPERA 5, 3 0, 0 FIGHT
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Solving the Game
The Battle of the Sexes
FIGHT OPERA P2
5, , 0
0, , 5
equity
FIGHT
OPERA
(0,0)
bargaining
power or
skill
efficiency
(1,1)
(5/8, 3/8) P1
NE = {(1, 1); (0, 0); (5/8, 3/8)}
Game 6.

31. Existence of Nash Equilibrium
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Existence of Nash Equilibrium
There can be (i) a single pure-strategy NE; (ii) a single mixed-strategy NE; or (iii) two pure-strategy NEs plus a single mixed-strategy NE (for x=z; y=w).
Prisoner’s Dilemma Button-Button Battle of the Sexes
GAME 3. GAME 2. GAME 6. q 1
(i) (ii) (iii) p

32. Properties SADDLEPOINT v. NASH EQUILIBRIUM
STABILITY: Is it self-enforcing?
YES YES
UNIQUENESS: Does it identify an unambiguous course of action?
YES NO
EFFICIENCY: Is it at least as good as any other outcome for all players?
--- (YES) NOT ALWAYS
SECURITY: Does it ensure a minimum payoff?
EXISTENCE: Does a solution always exist for the class of games? YES YES

33. Properties
[I]f game theory is to provide a unique solution to a game-theoretic problem then the solution must be a Nash equilibrium in the following sense. Suppose that game theory makes a unique prediction about the strategy each player will choose. In order for this prediction to be correct, it is necessary that each player be willing to choose the strategy predicted by the theory. Thus each player’s predicted strategy must be that player’s best response to the predicted strategies of the other players. Such a prediction could be called strategically stable or self-enforcing, because no single player wants to deviate from his or her predicted strategy (Gibbons: 8).

34. Properties
Normative theories tell us how a rational player will behave.
Descriptive theories tell us how real people actually behave.
Prescriptive theories offer advice on how real people should behave.

35. Problems Indeterminacy: Nash equilibria are not usually unique.
2. Inefficency: Even when they are unique, NE are not always efficient.

36. Problems
T T2
Multiple and Inefficient Nash Equilibria S1 S2
5, ,1
1, ,3
When is it advisable to play a prudent strategy in a nonzero-sum game?
What do we need to know/believe about the other player?

37. Problems
T T2
Multiple and Inefficient Nash Equilibria S1 S2
5, ,1
1, ,3
When is it advisable to play a prudent strategy in a nonzero-sum game?
What do we need to know/believe about the other player?