Microeconomics Theory - Lecture Notes Spring 2006 - Economics, Study notes of Economics

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Dirk Bergemann

Department of Economics

Yale University

Game Theory and Information Economics

January 2006

Springer-Verlag

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Part III. Static Games of Incomplete Information

• 1. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 1.1 Game theory and parlor games - a brief history
  • 1.2 Game theory in microeconomics
• 2. Normal Form : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Part I. Static Games of Complete Information
  • 2.1 Leading Examples
  • 2.2 The Normal Form Representation
  • 2.3 Rational Strategic Behavior
    • 2.3.1 Dominant Strategies
    • 2.3.2 Iterated Deletion of Strictly Dominated Strategies:
• 3. Nash Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 3.1 Best Response Correspondences
  • 3.2 Mixed Strategies
  • 3.3 Existence of Nash Equilibrium:
  • 3.4 Imperfect Competition
    • 3.4.1 An Existence Problem
    • 3.4.2 Reconciling quantity and price competition
    • 3.4.3 Imperfect Substitutes: Monopolistic Competition and the Dixit/Stiglitz model
  • 3.5 Entry and the Competitive Limit 12.E, 12.F
    • 3.5.1 Competitive Case MWG 10.F
    • 3.5.2 Modelling Entry 12.E
    • 3.5.3 The Competitive Limit 12.F
• 4. Perfect Information Games: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Part II. Dynamic Games of Complete Information
  • 4.1 Extensive (Tree) Form to Normal Form
    • 4.1.1 Nash Equilibria
    • 4.1.2 Backward Induction and Credible Threats: G 2.1.A, 2.1.D; MWG 9.B
  • 4.2 The Extensive Form Representation
  • 4.3 Subgame Perfection:
  • 4.4 Bargaining
  • 4.5 Nash Bargaining Problem: MWG 22.E
    • 4.5.1 The \Nash Program": Alternating O ers and the Nash Bargaining Solution
• 5. Repeated Games and Folk Theorems: : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 4 Contents
  • 5.1 In nitely Repeated Games
  • 5.2 Folk Theorems
• 6. Sequential Rationality : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Part IV. Dynamic Games of Incomplete Information
  • 6.1 Introduction Part V. Information Economics
• 7. Akerlof 's Lemon Model : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 7.1 Basic Model
  • 7.2 Extensions
  • 7.3 Wolinsky's Price Signal's Quality
  • 7.4 Conclusion
  • 7.5 Reading
• 8. Job Market Signalling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 8.1 Pure Strategy
  • 8.2 Perfect Bayesian Equilibrium
  • 8.3 Equilibrium Domination
  • 8.4 Informed Principal
    • 8.4.1 Maskin and Tirole's informed principal problem
  • 8.5 Spence-Mirrlees Single Crossing Condition
    • 8.5.1 Separating Condition
  • 8.6 Supermodular
  • 8.7 Supermodular and Single Crossing
  • 8.8 Signalling versus Disclosure
  • 8.9 Reading
• 9. Moral Hazard : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 9.1 Introduction and Basics
  • 9.2 Binary Example
    • 9.2.1 First Best
    • 9.2.2 Second Best
  • 9.3 General Model with nite outcomes and actions
    • 9.3.1 Optimal Contract
    • 9.3.2 Monotone Likelihood Ratio
    • 9.3.3 Convexity
  • 9.4 Information and Contract
    • 9.4.1 Informativeness
    • 9.4.2 Additional Signals
  • 9.5 Linear contracts with normally distributed performance and exponential utility
    • 9.5.1 Certainty Equivalent
    • 9.5.2 Rewriting Incentive and Participation Constraints
      • Contents
  • 9.6 Readings
• 10. Introduction : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Part VI. Mechanism Design
• 11. Adverse selection: Mechanism Design with One Agent : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 11.1 Monopolistic Price Discrimination with Binary Types
    • 11.1.1 First Best
    • 11.1.2 Second Best: Asymmetric information
  • 11.2 Continuous type model
    • 11.2.1 Information Rent
    • 11.2.2 Utilities
    • 11.2.3 Incentive Compatibility
  • 11.3 Optimal Contracts
    • 11.3.1 Optimality Conditions
    • 11.3.2 Pointwise Optimization
• 12. Mechanism Design Problem with Many Agents : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 12.1 Introduction
  • 12.2 Model
  • 12.3 Mechanism as a Game
  • 12.4 Second Price Sealed Bid Auction
• 13. Dominant Strategy Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
• 14. Bayesian Equilibrium : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 14.1 First Price Auctions
  • 14.2 Optimal Auctions
    • 14.2.1 Revenue Equivalence
    • 14.2.2 Optimal Auction
  • 14.3 Additional Examples
    • 14.3.1 Procurement Bidding
    • 14.3.2 Bilateral Trade
    • 14.3.3 Readings
• 15. Eciency : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 15.1 First Best
  • 15.2 Second Best
• 16. Social Choice : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
  • 16.1 Social Welfare Functional
  • 16.2 Social Choice Function

1. Introduction

Game theory is the study of multi-person decision problems. The focus of game theory is interdependence, situations in which an entire group of people is a ected by the choices made by every individual within that group. As such they appear frequently in economics. Models and situations of trading processes (auction, bargaining) involve game theory, labor and nancial markets. There are multi-agent decision problems within an organization, many person may compete for a promotion, several divisions compete for investment capital. In international economics countries choose tari s and trade policies, in macroeconomics, the FRB attempts to control prices. Why game theory and economics? In competitive environments, large populations interact. How- ever, the competitive assumption allows us to analyze that interaction without detailed analysis of strategic interaction. This gives us a very powerful theory and also lies behind the remarkable property that ecient allocations can be decentralized through markets. In many economic settings, the competitive assumption does not makes sense and strategic issues must addressed directly. Rather than come up with a menu of di erent theories to deal with non-competitive economic environments, it is useful to come up with an encompassing theory of strategic interaction (game theory) and then see how various non-competitive economic environments t into that theory. Thus this sec- tion of the course will provide a self-contained introduction to game theory that simultaneously introduces some key ideas from the theory of imperfect competition.

1. What will each individual guess about the other choices?
2. What action will each person take?
3. What is the outcome of these actions?

In addition we may ask

1. Does it make a di erence if the group interacts more than once?
2. What if each individual is uncertain about the characteristics of the other players?

Three basic distinctions may be made at the outset

1. non-cooperative vs. cooperative games
2. strategic (or normal form) games and extensive (form) games
3. games with perfect or imperfect information

In all game theoretic models, the basic entity is a player. In noncooperative games the individual player and her actions are the primitives of the model, whereas in cooperative games coalition of players and their joint actions are the primitives.

1.1 Game theory and parlor games - a brief history

1. 20s and 30s: precursors a) E. Zermelo (1913) chess, the game has a solution, solution concept: backwards induction

8 1. Introduction

b) E. Borel (1913) mixed strategies, conjecture of non-existence

2. 40s and 50s: core conceptual development a) J. v. Neumann (1928) existence in of zero-sum games b) J. v. Neumann / O. Morgenstern (1944) Theory of Games and Economic Behavior: Axiomatic expected utility theory, Zero-sum games, cooperative game theory c) J. Nash (1950) Nonzero sum games
3. 60s: two crucial ingredients for future development: credibility (subgame perfection) and incomplete information a) R. Selten (1965,75) dynamic games, subgame perfect equilibrium b) J. Harsanyi (1967/68) games of incomplete information
4. 70s and 80s: rst phase of applications of game theory (and information economics) in applied elds of economics
5. 90s and on: real integration of game theory insights in empirical work and institutional design

 For more on the history of game theory, see Aumann's entry on \Game Theory" in the New Palgrave Dictionary of Economics.

1.2 Game theory in microeconomics

1. decision theory (single agent)
2. game theory (few agents)
3. general equilibrium theory (many agents)

2. Normal Form

A game is a formal representation of a situation in which a number of individuals interact in a setting with strategic interdependence.. The welfare of an agent depends not only on his action but on the action of other agents. The degree of strategic interdependence may often vary.

Example 2.0.1. Monopoly, Oligopoly, Perfect Competition

To describe a strategic situation we need to describe the players, the rules, the outcomes, and the payo s or utilities.

2.1 Leading Examples

Example 1: (Duopoly). Two rms; constant marginal cost: $1; no xed cost; total demand curve: Q = 13P Example 2: (Partnership). Two partners; cost of e ort: 4; output per partner making e ort: 6. Output split 50=50. Example 3: (Sealed Bid Second Price Auction). Two bidders; i's reservation value is vi. Highest bidder pays second highest bid. (If they bid the same, each has a 12 chance of getting the prize and paying the (equal) bid).

2.2 The Normal Form Representation

Each example entailed \players" making simultaneous decisions. Each example is strategic, that is, each player's \utility" depends on the actions of others. We want a general language of rational strategic behavior in which we can describe each of the examples. But rst what is a game? It is a set of players:

I = f 1 ; 2 ; :::; Ig ; (2.1)

a set of possible strategies for each player

8 i; si 2 Si; (2.2)

where each individual player i has a set of pure strategies Si available to him and a particular element in the set of pure strategies is si 2 Si. Finally there are payo -o functions for each player i:

ui : S 1  S 2 


 SI! R. (2.3)

A pro le of pure strategies for the players is given by

s = (s 1 ; :::; sI ) 2

I  i=

Si

or alternatively by separating the strategy of player i from all other players, denoted by i:

2.3 Rational Strategic Behavior 13

E ort No E ort E ort 2,2 -1, No E ort 3,-1 0, Whatever action 2 chooses, 1's best action is to choose no e ort. We say \no e ort" is dominant strategy. Notice that in example 2, each player has a dominant strategy (no e ort); but when players choose their dominant strategies, the outcomes are inecient. In particular, if both exert e ort, both players are better o. Thus even in these simplest type of examples, rationality fails to imply eciency. We will need some more precise de nitions of domination.

 Some Notation:

A typical strategy pro le is s = (s 1 ; :::; si 1 ; si; si+1; :::; sI ) Write si = (s 1 ; :::; si 1 ; si+1; :::; sI ) for a vector specifying strategies for all players except player i. Write Si for the set of such pro les, i.e. Si = S 1  :::  Si 1  Si+1  :::  SI. Write (s^0 i; si) for a strategy pro le where i chooses s^0 i and all other players choose according to s.

De nition 2.3.1. Strategy si strictly dominates s^0 i if

ui (si; si) > ui (s^0 i; si) , for all si 2 Si

De nition 2.3.2. Strategy si is strictly dominant if si strictly dominates s^0 i for all s^0 i 6 = si.

De nition 2.3.3. Strategy si dominates s^0 i if

ui (si; si)  ui (s^0 i; si) , for all si 2 Si and ui

si; s^0 i

ui

s^0 i; s^0 i

, for some s^0 i 2 Si

De nition 2.3.4. Strategy si is dominant if si dominates s^0 i for all s^0 i 6 = si.

Thus - by de nition - if si strictly dominates s^0 i, si dominates s^0 i. If si is strictly dominant, then s^0 i is dominant.

Let's check for dominant strategies in the examples. Example 2: (Partnership). \No e ort" is strictly dominant, so \No e ort" is dominated. Example 3: (Sealed Bid Second Price Auction). There is no strictly dominant strategy. The strategy si = vi is a dominant strategy for each player. We check for player 1. Recall that

u 1 (s 1 ; s 2 ) =

v 1 s 2 , if s 1 > s 2 1 2 (v^1 ^ s^2 ) , if^ s^1 =^ s^2 0, if s 1 < s 2 If s 2  v 1 , u 1 (v 1 ; s 2 ) = 0  u 1 (s 1 ; s 2 ) for all s 1 2 R+. So v 1 gives at least as much as any other strategy, if s 2  v 1. If s 2 < v 1 , u 1 (v 1 ; s 2 ) = v 1 s 2 , so

u 1 (v 1 ; s 2 ) u 1 (s 1 ; s 2 ) =

v 1 s 2 , if s 1 < s 2 1 2 (v^1 ^ s^2 ) , if^ s^1 =^ s^2 0, if s 1 > s 2

 0 for all s 1 2 R+

Since this is non-negative for all s 1 , v 1 does at least as well as any other strategy if s 2 < v 1. To show that v 1 is a dominant strategy we must also check that for all s 1 6 = v 1 , there exists s 2 such that u 1 (v 1 ; s 2 ) > u 1 (s 1 ; s 2 ). Consider rst the case where s 1 > v 1 ; now if v 1 < s 2 < s 1 , u 1 (s 1 ; s 2 ) = v 1 s 2 < 0 = u 1 (v 1 ; s 2 ). On the other hand, suppose s 1 < v 1 ; now if s 1 < s 2 < v 1 , then u 1 (s 1 ; s 2 ) = 0 < v 1 s 2 = u 1 (v 1 ; s 2 ). Thus v 1 is a dominant strategy for player 1. In fact, it is the only dominant strategy for player 1. It is clearly not strictly dominant: if s 2  v 1 , any strategy s 1 with s 1  s 2 gives the same optimal payo of 0.

14 2. Normal Form

[lecture 2:]

De nition 2.3.5. Fix strategy pro le s. If each si is a dominant strategy for player i, then s is a dominant strategies equilibrium.

Example 1: (Duopoly). Suppose that s 1 + s 2 < 12. Then u 1 (s 1 ; s 2 ) = s 1 (12 s 1 s 2 ) (Check!). Now du 1 ds 1 = 12^ ^ s^2 ^2 s^1. Thus at an interior maximum, we have 12^ ^ s^2 ^2 s^1 = 0, i.e.,^ s^1 = 6^ ^

1 2 s^2. In particular, if s 2 2 [0; 12), u 1

6 12 s 2 ; s 2

u 1 (s 1 ; s 2 ) for all s 1 6 = 6 12 s 2. This is the more typical case: for every action of player 2, player 1 has a di erent best action.

2.3.2 Iterated Deletion of Strictly Dominated Strategies:

Example 4:

Left Middle Right Up 1,0 1,2 0, Down 0,3 0,1 2,

\Middle" strictly dominates \Right". But \Middle" does not strictly dominate \Left" and \Left" does not strictly dominate \Middle", so the \column player" does not have a strictly dominant strategy. Nor does the \row player". But since \Right" is strictly dominated by some other strategy (\Middle"), a rational row player will not expect the column player to choose it. Thus we get:

Left Middle Up 1,0 1, Down 0,3 0,

But \Down" is strictly dominated in this game, so...

Left Middle Up 1,0 1,

\Left" is strictly dominated in this game, so...

Middle Up 1,

This process is known as iterated deletion of strictly dominated strategies. (Up, Middle) is the unique strategy pro le which survives iterated deletion of strictly dominated strategies.

De nition 2.3.6 (Iterated Strict Dominance). The process of iterated deletion of strictly dominated strategies proceeds as follows: Set S^0 i = Si. De ne Sni recursively by

Sni =

si 2 S in ^1 @s^0 i 2 S in ^1 , s.th. ui (s^0 i; si) > ui (si; si) , 8 si 2 Sn i 1 ;

Set

S^1 i =

^1

n=

Sni :

The set S i^1 is then the set of pure strategies that survive iterated deletion of strictly dominated strategies.

3. Nash Equilibrium

A Nash equilibrium is a pro le of actions were each player's action is optimal given the actions of others. Formally:

De nition: Strategy pro le s^ is a Nash equilibrium if, for all i = 1; :::; I and all si 2 Si,

ui

s i ; si

 ui

si; si

Example 5: (Coordination Failure). (Invest, Invest) and (Don't Invest, Don't Invest) are both Nash equi- libria. Exercise: If each s i is a dominant strategy, then s^ is a Nash equilibrium.

We have shown that this gives us a Nash equilibrium of examples 2 and 3.

Exercise: If s^ is the unique strategy pro le surviving iterated deletion of strictly dominated strategies, then s^ is the unique Nash equilibrium.

We have shown that this gives us a Nash equilibrium in example 4. As an exercise, you can show that it gives a Nash equilibrium in example 1.

3.1 Best Response Correspondences

De nition: Write i (si) for player i's best response(s) to si. Thus:

i (si)  arg max si 2 Si

ui (si; si)

 fsi 2 Si : ui (si; si)  ui (s^0 i; si) for all s^0 i 2 Sig

Let  (s)  fs^0 2 S : s^0 i 2 i (si) for each ig. Now s^ is a Nash equilibrium if and only if s^2  (s).

Example 6: (Contribution to a public good). Two individuals, individual i has income w and chooses gi 2 [0; w], contribution to public good. Individual's private consumption w gi. His utility is ui (g 1 ; g 2 ) = ln (w gi) + (1 ) ln (g 1 + g 2 ), where 2 (0; 1). Now (assuming interior solution) we have at maximum: du 1 dg 1

w g 1

g 1 + g 2

i.e. (1 ) w (1 ) g 1 = g 1 + g 2 so  1 (g 2 ) = (1 ) w g 2. (Technically,  1 (g 2 ) = f(1 ) w g 2 g). Similarly,  2 (g 1 ) = (1 ) w g 1. Plotting best responses (see gure 1), we see g 1 = g 2 = g^ is unique Nash equilibrium, where g^ = (1 ) w g, i.e. g^ = (1 1+^ )w. What is the ecient symmetric level of contribution? Choose g to maximize (1 ) ln (2g) + ln (w g) i.e. set 1 ge (^) wge = 0, i.e. set (1 ) (w ge) = ge^ i.e. ge^ = (1 ) w > (1 1+^ )w.

3.3 Existence of Nash Equilibrium: 19

Dominated Strategies Revisited:. Consider the following game:

Left Right Up 4,0 -2, Middle 0,0 0, Down -2,0 4,

Pure strategy \Middle" is not dominated by any pure strategy. But \Middle" is strictly dominated by the strategy putting probability 12 on Left and 12 on Right.

3.3 Existence of Nash Equilibrium:

Theorem (Nash 1950) Every nite action game has at least one Nash equilibrium.

Proof. De ne : !  as follows. Each mixed strategy can be thought of as a vector and Euclidean distance between mixed strategy vectors can be described in the usual way:

k^0 i ik =

s X

si 2 Si

(^0 i (si) i (si))^2

Let vi (^0 i; ) = ui (^0 i; i) c k^0 i ik 2 (where c > 0) i () =^ arg max ^0 i 2 i

vi (^0 i; )

() = f (^) i ()gIi=

Interpretation: is a \better response" function with quadratic adjustment costs.

1.  is non-empty, compact and convex.
2. vi is strictly concave in ^0 i ( rst term is linear, and thus concave, second term is negative quadratic, and thus strictly concave). Thus (^) i is uniquely de ned and continuous in . Thus is a continuous function on .
3. If ^ = (), then ^ is a Nash equilibrium. Suppose not. Then there exists i and i 2 i such that
   = ui

i; i

ui

 i ; i

1. Now:

vi ("i + (1 ")  i ; ) vi ( i ; ) = "
c"^2 ki  i k^2

which is strictly positive for " suciently close to zero.

Brouwer's Fixed Point Theorem: Suppose X is a non-empty, compact, convex subset of RN^ and f : X! X is continuous. Then f has a xed point, i.e., there exists x 2 X with x = f (x). Now Nash equilibrium exists by points 1 through 3 above.

 This proof follows Geanakoplos (1996): \Nash and Walras Equilibrium Via Brouwer," Cowles Foundation Discussion Paper #1131.

Existence with continuous strategy spaces: If payo s are continuous, then existence is no problem. With discontinuous payo s, existence is a real problem. Best responses may not be well de ned. Discontinuous payo s arise naturally in economic settings, and cause real problems. They arise naturally in price-setting games (to be discussed furthers in the next section). We return to this issue at the end of the next section.

20 3. Nash Equilibrium

3.4 Imperfect Competition

Homogenous Good. Assume constant marginal cost c > 0 (with varying marginal cost, entry as- sumptions become crucial; we will return to this later). Demand x (p); assume di erentiable, non- increasing, strictly decreasing if x (p) > 0, x (p) = 0 for all p  p. This implies inverse demand function p (q) = min fp : q = x (p)g.

Competitive Case. Choose q to maximize q [p c]. Must have pc^ = c, qc^ implicitly de ned by

p (qc) = c

Monopoly. Choose p to maximize pro ts: x (p) [p c] Choose q to maximize pro ts: q [p (q) c] First Order Condition:

Marginal Revenue = p (q) + qp^0 (q) = c = Marginal Cost

Equivalently,

p c p

qp^0 (q) p

where  is the own price elasticity of demand. Let qm^ solve

p (qm) + qmp^0 (qm) = c

and

pm^ = p (qm)

Observe that pm^ > pc^ and qm^ < qc.

Oligopoly. I rms

Quantity Competition (\Cournot Competition"). Payo functions:

i (q 1 ; :::::; qI ) = qi

(^4) p

X^ I

j=

qj

A (^) c

First order condition:

di dqi

= p

X^ I

j=

qj

A (^) + qip^0

X^ I

j=

qj

A (^) c = 0

In a symmetric equilibrium, qi = q for all i (by continuity, a symmetric solution to the rst order condition will exist). So if we write qI^ = Iq for total quantity produced in oligopoly, we have qI^ solving

p

qI^

I

qI^ p^0

qI^

= c.

Thus qm^ = q^1 and qc^ = q^1. Exercise: how does qI^ vary as I increases....