APPLIED MICROECONOMICS - static.uni-graz.at

1

APPLIED MICROECONOMICS Christian Klamler

Slides based on: Pindyck R.S. and D.L. Rubinfeld (2009): “Microeconomics”, 7th edition, Pearson International Edition (and on slides by Companion Webpage, Pearson Education). Osborne, M.J. (2004): “An Introduction to Game Theory”, Oxford University Press. Gibbons, R. (1992): “A Primer in Game Theory”, Harvester Wheatsheaf, New York. This work is protected by regional copyright laws and is provided solely for the use of instructors in teaching their courses and assessing student learning. Dissemination or sale of any part of this work (including on the Internet) will destroy the integrity of the work and is not permitted.

© Osborne, Pearson Int. and C. Klamler

2 General Information n  Time and location: see UniGrazOnline for details n  Course material

¨  available on my webpage (access via usual uni-graz login) ¨  [email protected]

n  Office hours ¨  Mon 13.30 – 15.00

n  Grading: 3 exams or short tests ¨  18 Oct (max 15 points) ¨  15 Nov (max 40 points) ¨  Fr. 16 Dec (max 45 points); HS 15.04; 10am

n  in-class participation n  more than 50 points necessary for positive grade!


3 Content of course Lecture and slides are based on chapters from the following textbooks: n  Gibbons, R. (1992): A Primer in Game Theory, Harvester

Wheatsheaf, New York. ¨  Chapters 1 – 2; Static and dynamic games of complete information

n  Osborne, M.J. (2004): An Introduction to Game Theory, Oxford University Press. ¨  Chapters 2 – 6; Strategic games and extensive games

n  Pindyck R.S. and D.L. Rubinfeld (2003 or younger): Microeconomics, 5th (or higher) edition, Pearson Education. ¨  Chapter 13 – Game Theory

n  Strategic aspects in decision making ¨  Chapter 17 – Asymmetric information

n  Incomplete information as market failure ¨  Chapter 18 – Externalities and Public Goods

n  Market failures and government intervention

n  prerequisite: VU Microeconomics!!

4

GAME THEORY


5 Introduction

What is game theory? ¨  Tool to analyse situations in which decision-makers

interact. ¨  Wide range of applications (not only “real” games)

n  economics ¨  firms competing for business ¨  bidders competing in an auction

n  political science ¨  political candidates competing for votes

n  law ¨  jury members deciding on a verdict

n  biology ¨  animals fighting over prey

n  etc.


6 Introduction – Example

n  Firms competing for business n  each firm controls its own price

¨  but not the price of the other firms! n  each firm cares about all the firms’ prices, because these prices

affect its sales ¨  should be considered in its own pricing decision

n  which price should a firm set?

Those and similar questions will be discussed in this course!


7 Introduction – Historical aspects

n  Antoine-Augustine Cournot n  John von Neumann; Oskar Morgenstern n  John Nash n  Nobel-Prize 1994: Harsanyi, Nash, Selten n  Nobel-Prize 2005: Aumann, Schelling n  Nobel-Prize 2007: Hurwicz, Maskin, Myerson n  Nobel-Prize 2012: Roth, Shapley


8 Introduction – Basic Concepts

Modelling decision-makers (players)

n  Players choose actions/strategies from a set of possible actions ¨  what could those actions be?

n  Players have preferences over states of the world ¨  complete, consistent (transitive)

n  Theory of rational choice says ¨  that the action chosen by a decision-maker is at least as good, according

to her preferences, as every other available action.

n  Difference between ¨  consumer theory ¨  game theory


9 Introduction – Basic Concepts

n  relationship between players ¨  cooperative vs. non-cooperative games

n  binding contracts

n  structure of the game ¨  simultaneous vs. sequential moves

n  informational aspects ¨  common knowledge

n  information or events that all players know, everybody knows that everybody knows them, everybody knows that everybody knows that everybody knows them, etc.

¨  complete information n  each player’s payoff function is common knowledge

¨  perfect information n  at each move, the player knows the full history of the game so far

10

Strategic / Normal Form / Static Games


11 Definitions

n  What is a game?

n  A strategic game (normal form game) consists of ¨  a set of players, N ¨  for each player i a set of actions, Ai={ai,bi,…}

¨  for each player, preferences over the set of action profiles n  e.g. represented by utility function ui(.)

¨  important: players move simultaneously!


12 Example – Prisoners’ Dilemma Story:

¨  2 suspects in a major crime are held in separate cells n  Enough evidence to convict each of them of a minor offence n  Not enough evidence to convict both of them of the major offence unless one

of them acts as an informer (to fink) ¨  They are both independently faced with the following decision problem

n  if neither confesses, both get 2 years n  if only one confesses he gets 1 year, the other 10 years n  if both confess, both get 5 years

n  Formal representation ¨  N={1,2} ¨  A1={fink, quiet}=A2 ¨  u1(f,f)=-5; u1(f,q)=-1; u1(q,f)=-10; u1(q,q)=-2 ¨  u2(f,f)=-5; u2(f,q)=-10; u2(q,f)=-1; u2(q,q)=-2


13 Payoff matrix – Prisoners’ dilemma

¨  N={1,2} ¨  A1={fink, quiet}=A2 ¨  u1(f,f)=-5; u1(f,q)=-1; u1(q,f)=-10; u1(q,q)=-2 ¨  u2(f,f)=-5; u2(f,q)=-10; u2(q,f)=-1; u2(q,q)=-2

players

actions

actions

payoffs

let us transform this information into a matrix


14 Prisoners‘ dilemma

n  Dominant Strategy ¨  it always leads to a higher payoff, irrespective of what action the

other player chooses. ¨  what happens if both follow their dominant strategies? ¨  Pareto (in)efficiency

n  outcome is Pareto efficient if it is not possible to improve one player‘s payoff without lowering the other player‘s payoff

n  Pareto improvement

n  What will the (rational) suspects do?


15 Dominant strategies – formal definitions n  Notation

¨  a = (a1,…,an) is a strategy profile, i.e. a strategy ai for each player i. ¨  (a‘i,a-i) means that all j≠i choose their strategy according to profile

a, while player i chooses strategy a‘i ¨  ui(.) is the utility function of player i, which attaches to each

strategy profile the utility that i derives from it.

Strategy ai‘‘ strictly dominates strategy ai‘, if

for all strategy profiles a-i of the other players. Strategy ai‘ is called strictly dominated.

n  Would you ever want to play a strategy that is strictly dominated? n  Best response function


16 Iteration

n  A strictly dominated strategy is NEVER a best response ¨  no rational belief of a player could ever assume another player to

choose a strictly dominated strategy n  given common knowledge that the players are rational

¨  strictly dominated strategy can be eliminated n  changes the structure of the payoff matrix n  repeat the process

¨  iterated elimination of strictly dominated actions

equilibrium in dominant strategies


17 Exercises


18 Dominant strategies

n  Ex.: Do we have strictly dominant or strictly dominated strategies here?

Strategy ai‘‘ weakly dominates strategy ai‘, if

for all strategy profiles a-i of the other players, and

for some strategy profile a-i. Strategy ai‘ is called weakly dominated.


19 Iteration

n  What about weakly dominated actions? ¨  those can be best responses!

n  less appeal

¨  solution depends on the order of elimination


20 Solutions n  What if there are no dominant strategies?

¨  can we explain behaviour or predict outcome? ¨  best strategy for any given player depends – in general – on the

other players’ actions n  rational players n  complete information n  common knowledge

In case game theory makes a (unique) prediction about the strategy each player should choose, for this prediction to be correct, the players need to be willing to play the predicted strategy.

best response strategically stable self-enforcing


21 Nash equilibrium

A Nash equilibrium is an action profile a* with the property that no player i can do better by choosing an action different from ai*, given that every other player j adheres to aj*.

A Nash equilibrium somehow corresponds to a steady state. There is no individual incentive to choose a different strategy. Hence, it embodies a stable “social norm”: if everyone else adheres to it, no individual wishes to deviate from it.

John Nash


22 Nash equilibrium – formal definitions

The strategy profile a* is a Nash equilibrium if, for every player i and every action ai of player i, a* is at least as good according to player i’s preferences as the action profile (ai,a-i*) in which player i chooses ai while every other player j chooses aj*. Equivalently, for every player i,

for each strategy ai of player i.

n  Notation ¨  a = (a1,…,an) is a strategy profile, i.e. a strategy ai for each player i. ¨  (a’i,a-i) means that all j≠i choose their strategy according to profile

a, while player i chooses strategy a’i ¨  ui(.) is the utility function of player i, which attaches to each

strategy profile the utility that i derives from it.


23 Nash equilibrium and best responses

n  We can also define a Nash equilibrium in terms of best response functions:

The strategy profile a* is a Nash equilibrium if and only if every player’s strategy is a best response to the other players’ strategies, i.e. for all players i:


24 Exercises


25 Nash equilibrium: Prisoners’ Dilemma

n  many situations structurally similar to PD ¨  2 individuals work on a joint project ¨  each of them can work hard (h) or not (n) ¨  preferences are assumed as follows

¨  based on those preferences, certain payoff functions could lead to the following payoff matrix:

n  similar conclusions as in the PD-game n  free-rider behaviour n  other examples: duopoly, common properties, etc.


26 Exercises

n  Is the structure of the following models different from the one in the PD-game? If so, in what form?

n  Which numbers do you need to insert, such that the payoff matrix becomes structurally equivalent to the one in the PD-game?


27 Nash equilibrium: Battle of Sexes

n  In the PD-game, main question is whether the players will cooperate. n  In the following game the players agree that it is better to cooperate

than not to cooperate, but they disagree about the best outcome. n  many structurally similar situations

n  The BoS game: ¨  2 people have to decide on where to go

n  2 options: concert (C) or football (F) n  one person prefers C, the other F n  both prefer going together over going alone


28 Nash equilibrium: Coordination Game n  adapt the BoS game by changing the payoff as follows:

n  Nash equilibria? ¨  what is the difference to the BoS game ¨  focal point

n  Thomas Schelling (1960) ¨  Nash‘s theory says nothing about the actual equilibrium that will

occur in case of several Nash equilibria. However, equilibria that are obviously better for everyone, seem to occur more often.


29 Nash equilibrium: Chicken game n  the following game makes coordination difficult

¨  players can either choose to back down (B) or challenge (C)

n  Nash equilibria? ¨  what is the difference to the BoS game

¨  compare non-Nash outcome (B,B) to Nash-equilibria ¨  chicken game has many applications

¨  e.g. military conflicts (arms races)


30 Nash equilibrium: Matching Pennies n  2 players choose – simultaneously – whether to show the head or

the tail of a coin ¨  If they show the same side, P2 pays P1 one Euro. ¨  If they show different sides, P1 pays P2 one Euro. ¨  What does the payoff matrix look like?

¨  As the players’ interests are diametrically opposed, such a game is strictly competitive and purely conflictual. (zero-sum game)

¨  No Nash equilibrium (at least in pure strategies)

31

Strategic Games - Illustrations

STRATEGIC GAMES ILLUSTRATIONS


32 International trade n  trade theory shows, that trade can be mutually beneficial

¨  still governments try to restrict imports by imposing tariffs n  see USA vs. EU trade conflicts

¨  various game theoretic applications possible ¨  countries decide between tariffs (T) and no tariffs (N)

n  Nash equilibria?


33 Cournot’s model of oligopoly

n  Model the competition between a small number of sellers (oligopoly) ¨  Cournot (1838) modeled the industry as the following strategic

game: n  Players: the firms n  Actions: each firm’s set of possible outputs n  Preferences: represented by each firm’s profit, given in its profit

function


34 Cournot’s model – duopoly n  what do the firms “think” about the situation?

¨  for any given action of firm 2, firm 1’s actions yield her various outcomes

¨  choose the “best action” for any of firm 2’s actions


35 Cournot’s model – duopoly n  Best response function

¨  or reaction curve ¨  can we find an equilibrium?


36 Cournot’s model – duopoly

n  How does the Nash equilibrium outcome compare with collusive outcomes ¨  firms could agree on size of production ¨  would the firms be better off?

¨  collusion could – in general – be of benefit to the firms n  will it occur?

lower profit for firm 2

lower profit for firm 1


37 Cournot’s model – duopoly n  Numerical example

n  P = 100 – Q n  C1(Q1) = 10Q1; C2(Q2) = 10Q2

¨  What does the Nash-Cournot equilibrium look like? ¨  What is the collusive outcome?

n  but what problems do occur with this outcome?

n  What happens if we increase the number of firms? ¨  keep the assumption of identical linear cost functions of the firms ¨  inverse demand function remains P(Q) ¨  what is the Nash equilibrium in this case

¨  the price in the NE of Cournot’s game decreases as the number of firms increases, approaching c


38 Bertrand’s model of oligopoly

n  Firms choose prices and not quantity as in Cournot’s model ¨  each firm then produces enough to meet the demand it faces, given

the prices chosen by all the firms ¨  should this not lead to the same results as in Cournot’s model?

n  Bertrand’s oligopoly game ¨  Players: firms ¨  Actions: set of possible prices ¨  Preferences: represented by the firms’ profits

n  what would you expect to happen in case of homogeneous outputs ¨  consumers buy always at the lower price


39 Bertrand’s model – duopoly

n  assume differentiated products ¨  the higher priced good does not loose all its demand ¨  example: the firms’ demand functions are assumed to be:

1 1 212 2Q P P= − +

2 2 112 2Q P P= − +

¨  costs: fixed cost of 20, no variable costs ¨  profit function:

¨  profit maximizing price to be determined via:

¨  best responses


40 Bertrand’s model – duopoly

n  What is the Nash equilibrium pair of prices? n  What is the collusive outcome? n  Are there individual incentives to stick to an agreement?

TABLE 12.3 Payoff Matrix for Pricing Game

Firm 2

Charge $4 Charge $6

Firm 1 Charge $4 $12, $12 $20, $4

Charge $6 $4, $20 $16, $16


41 Electoral competition n  Game theory is important to model the foundation for many

theories of political phenomena n  what determines the number of political parties? n  what determines the policies that political parties propose? n  how is the outcome of an election affected by the electoral system?

n  Simple model is a strategic game of the following form: ¨  Players: candidates ¨  Actions: (political) positions (usually numbers on a left-right line) ¨  Preferences: payoff relative to voting outcome


42 Electoral competition n  Candidates have certain positions on the line

n  Voters have preferences over the positions n  the closer the candidate is to a voter’s ideal position, the more the voter

likes this candidate n  single-peaked preferences

n  In this model, each candidate attracts the votes of all citizens whose favourite positions are closer to her position than to the position of any other candidate.


43 Electoral competition n  special position: median

¨  position m with the property that half of the voters’ favourite positions are at most m, and half of them at least m

n  Which position should a candidate (in case of 2 candidates only) try to occupy?

median

n  Try to be as close to the median as possible to win the election n  Hotelling-Downs model of electoral competition n  tendency to move towards median position


44 Electoral competition n  many structurally similar situations

n  competition in product characteristics n  where should I locate myself to sell drinks on the beach?

¨  Beach location game

n  consumers buy drinks from the seller closest to them n  if the median is in the middle and your competitor is at “A”, then try to

put yourself between median and “A” n  eventually both will be at the median n  clustering at the median – examples: supermarkets, car stores, etc.


45 Auctions

n  In an auction a good is sold to the party who submits the highest bid. ¨  used to allocate significant economic resources

n  licenses (radio, mobile phones, …) n  oil and gas exploration rights n  government bonds n  works of art n  eBay

¨  many forms of auctions n  English auction

¨  bids called out sequentially and known to players n  Dutch auction

¨  price lowered until a buyer stops n  sealed bid auction

¨  all bids are private and made simultaneously ¨  I don’t know the bids of the other players


46 Auctions n  Further distinction

¨  private value auction n  each bidder knows own individual valuation n  valuation might differ between bidders n  e.g. painting

¨  common value auction n  item has same value to all bidders n  bidders uncertain about precise value and their estimates differ n  e.g. oil fields, party games (glass of coins) n  winner’s curse

n  Two pricing options: ¨  first-price auction

n  sales price equal to the highest bid ¨  second-price auction (Vickrey auction)

n  sales price equal to the second highest bid n  why should this make sense?


47 Auctions – First-price sealed-bid n  Model:

¨  vi: player i’s private value attached to the object n  assumption: v1 > v2 > … > vn

¨  bi: player i’s submitted (sealed) bid ¨  p: price, at which the object is finally sold

n  First-price sealed-bid auction ¨  bidders do not know the others’ bids ¨  highest bidder wins and pays the price she bids

n  The first-price sealed-bid auction is the following strategic game ¨  Players: n ≥ 2 bidders ¨  Actions: the set of possible bids bi

¨  Preferences:


48 Auctions – First-price sealed-bid

n  Nash equilibria in a first-price sealed-bid auction? ¨  is telling the truth a Nash equilibrium?

n  (b1, … ,bn) = (v1, v2, v3, … , vn) ¨  what if the bids are (v1-ε,v2,…,vn)? ¨  telling the truth is weakly dominated by other bids ¨  many Nash equilibria


49 Auctions – Second-price sealed-bid

n  The second-price sealed-bid auction is the following strategic game ¨  Players: n ≥ 2 bidders ¨  Actions: the set of possible bids bi

¨  Preferences:

n  where p is the selling price (i.e. second-highest bid)

n  Nash equilibria in a second-price sealed-bid auction? ¨  is telling the truth, i.e. (b1, … , bn) = (v1, … , vn), a Nash equilibrium? ¨  how many Nash equilibria do you find?

¨  could there be a Nash equilibrium in which the player n receives the object?

¨  what happens if we think of the bidding events to unfold over time? n  certain equilibria seem to be unreasonable


50 Auctions – Second-price sealed-bid

¨  i.e. for any bid bi ≠ vi, player i’s bid vi is at least as good as bi, no matter what the other players bid, and is better than bi for some actions of the other players.

In a second-price sealed-bid auction a player’s bid equal to her valuation weakly dominates all her other bids.

© Osborne 2005

¨  a second-price auction has many Nash equilibria, but the equilibrium (b1, … ,bn) = (v1, … , vn) has weakly dominating actions for all players.

51

Mixed Strategies


52 Introduction n  Reconsider Matching Pennies Game

¨  no Nash equilibrium n  always incentive to deviate

¨  what strategy would you choose? n  probability distribution n  maximize expected utility/payoff

¨  use preferences over lotteries ¨  lottery P with 3 possible outcomes (a,b,c) and probabilities (pa,pb,pc),

and lottery Q with probabilities (qa,qb,qc) ¨  Player i prefers lottery P to Q if

¨  player decides according to her expected utility ¨  von Neumann – Morgenstern preferences ¨ Bernoully payoff function


53 Mixed Strategies – Definitions

n  A mixed strategy of a player in a strategic game is a probability distribution over the player’s actions.

n  a profile of mixed strategies is usually denoted by α n  αi(ai) is the probability that player i attaches to her playing action αi

n  A strategic game (with vNM preferences) consists of ¨  a set of players ¨  for each player, a set of actions ¨  for each player, preferences regarding lotteries over action

profiles that may be represented by the expected value of a (“Bernoulli”) payoff function over action profiles.


54 Matching Pennies

n  If P1 chooses “H” with prob. p, and P2 chooses “H” with prob. q, then P1 will get the following payoff:

probabilities by P2

q 1-q

probabilities by P1

p

1-p

¨  if q < 0.5, then 4q-2<0 and hence p = 0 is optimal for P1 n  but then q < 0.5 cannot be optimal for P2

¨  if q > 0.5, then 4q-2>0 and hence p = 1 is optimal for P1 n  but then q > 0.5 cannot be optimal for P2

¨  only steady state if q = 0.5 and p = 0.5


55 Matching Pennies – best response functions n  What do the best response functions look like in this game?

¨  if q < 0.5, then p = 0 is optimal for P1 ¨  if q > 0.5, then p = 1 is optimal for P1 ¨  if q = 0.5, then any p ∈ [0,1] is optimal for P1

B1

B2

1

1

q

p 0

0.5

0.5


56 Mixed Strategies – Equilibrium

The mixed strategy profile α* in a strategic game with vNM preferences is a (mixed strategy) Nash equilibrium if, for each player i and every mixed strategy αi of player i, the expected payoff to player i of α* is at least as large as the expected payoff to player i of (αi, α-i*) according to a payoff function whose expected value represents player i‘s preferences over lotteries. Equivalently, for each player i

where Ui(α) is a player i‘s expected payoff to profile α.

The mixed strategy profile α* is a mixed strategy Nash equilibrium if and only if αi* is in Bi(α-i*) for every player i.


57 Mixed strategies – BoS game

n  mixed strategies NE in addition to NE in pure strategies in a BoS game possible? ¨  P1’s expected payoff in playing “B” is: 2q + 0·(1-q) ¨  P1’s expected payoff in playing “S” is: 0·q + 1·(1-q) ¨  Hence P1 will play “B” as long as 2q > (1-q), what is the case for q >

1/3

probabilities by P1

p

1-p

probabilities by P2

q 1-q


58 Mixed strategies – BoS game

n  3 Nash equilibria, two in pure strategies and one in mixed strategies.

B1

1

1

q

p 0

1/3

2/3

B2


59 Mixed strategy Nash equilibrium – results

Every strategic game with vNM preferences in which each player has finitely many strategies has a mixed strategy Nash equilibrium.

A strictly dominated strategy is not used with positive probability in any mixed strategy Nash equilibrium.

n  Is there always a NE in a strategic game?

n  Can strictly dominated actions be used in a mixed strategy NE?

What is special in those games?


60 Exercises

n  Calculate pure- and mixed-strategy NE for the following games?

61

Extensive Form or Dynamic Games


62 Extensive form games – definitions n  Normal form games suppress the sequential structure of decision-

making ¨  we implicitly assumed that each decision-maker chooses her strategy once

and for all n  Extensive form games describe the sequential structure of decision-

making explicitly ¨  allow for studying situations in which each decision-maker is free to change

her mind as events unfold ¨  complete and perfect information

n  An extensive form game with perfect information has four components: ¨  set of players ¨  a set of sequences (terminal histories) ¨  a player function that assigns a player to every (non-terminal) sequence ¨  for each player, preferences over the set of terminal histories


63 Example – entry game

n  An incumbent (I) faces the possibility of entry by a challenger (C) n  C as a firm considering entry into an industry occupied by I n  C as politician competing for leadership of a party with current leader I n  C as an animal competing with another animal (I) for leadership in group

n  How does the extensive game work? ¨  At the start of the game and after any (non-terminal) sequence of

events, a player chooses an action ¨  in the entry game, C has the options “in” and “out”

n  those actions start the game ¨  I has the option “acquiesce” and “fight”

n  those actions follow a previous move “in” of C


64 Example – entry game

n  N={I,C} n  Terminal histories

n  (in, acquiesce) n  (in, fight) n  (out)

n  player function P n  P(∅) = C n  P(in) = I

n  preferences n  uI(out) = 2; uI(in, acquiesce) = 1; uI(in, fight) = 0 n  uC(out) = 1; uC(in, acquiesce) = 2; uC(in, fight) = 0


65 Game tree

n  The set of all actions available to the player who moves after h is: A(h) = {a: (h,a) is a history} ¨  Example:

n  histories are ∅, in, out, (in, acquiesce), (in, fight) n  available actions at h:

¨  A(∅)={in, out} ¨  A(in)={acquiesce, fight}

C

I

in out

acq. fight

2,1 0,0

1,2

decisions taken at those points (histories)


66 Solving games in extensive form

n  consider questions of credibility n  backward induction

¨  not always well-defined n  length of terminal histories n  payoffs at decision nodes

C

Iin out

acq. fight

2,1 0,0

1,2

Which action would a potential challenger (C) choose in this game?


67 Strategies n  What is a strategy of player i in an extensive game with perfect

information? ¨  a function that assigns to each history h after which it is player i‘s

turn to move (i.e. P(h) = i) an action in action set Ai(h).

1

2

C D

E F

2,1 3,0 1,3 0,2

HG2

n  player 1: C, D n  player 2: EG, EH, FG, FH

each strategy is a contingency plan.


68 Strategies

n  what are the strategies for the two players? ¨  player 2 has 4 available strategies: which?

¨  Attention: A strategy of any player i specifies an action for every history after which it is player i’s turn to move, even for histories that, if the strategy is followed, do not occur.

n  e.g. DG und DH in the above example

1

2

C D

E F 2,0

3,1

0,0 1,2

HG1


69 Outcomes

n  A strategy profile determines the terminal history that occurs ¨  Let s be the strategy profile and P the player function ¨  Follow the strategies until a terminal history is reached.

n  denote this terminal history of s with O(s) n  O(s) is a list of actions

1

2C D

E F 2,0

3,1

0,0 1,2

H G 1

Determine O(s) for s = (DG,E)? Determine O(s) for s = (CH,E)?

Determine O(s) for s = (CH,F)?

Determine O(s) for s = (CG,E)?


70 Nash equilibrium

n  As in normal form games, we are interested in equilibria ¨  steady state

The strategy profile s* in an extensive game with perfect information is a Nash equilibrium if, for every player i and every strategy ri of player i, the terminal history O(s*) is at least as good according to player i‘s preferences as the terminal history O(ri,s-i*). Equivalently, for each player i,

for every strategy ri of i.

n  How can we find the Nash equilibria in an extensive game? ¨  transform extensive form game into normal form game

The set of Nash equilibria of any extensive game with perfect information is the set of Nash equilibria of it‘s strategic (or normal) form.


71 Normal form vs. extensive form

n  player C’s strategies are “in” and “out”

n  player I’s strategies are “acquiesce” and “fight”

n  leads to the following corresponding game in normal form:

C

Iin out

acq. fight

2,1 0,0

1,2

n  What are the Nash equilibria? n  How can we interpret them? n  What can be done?


72 Subgame

n  The notion of Nash equilibrium ignores the sequential structure of an extensive game

n  treats strategies as choices made once and for all before play begins n  can lead to Nash equilibria which are not robust

n  The following notion of equilibrium models a robust steady state ¨  requires each player’s strategy to be optimal, given the other players’

strategies n  not only at the start of the game n  but also after every possible history

n  Subgame ¨  For any nonterminal history h, the subgame following h is the part of

the game that remains after h has occurred. ¨  proper subgame ¨  number of subgames is equal to the number of nonterminal histories


73 Subgames 1

2C D

E F 2,0

3,1

0,0 1,2

HG 1

Determine the proper subgames of this game?

0,0 1,2

HG 1

2E F

3,1

0,0 1,2

HG 1


74 Subgame perfect equilibrium

A subgame perfect equilibrium is a strategy profile s* with the property that in no subgame can any player i do better by choosing a strategy different from si*, given that every other player j adheres to sj*.

C

Iin out

acq. fight

2,1 0,0

1,2

n  Nash equilibrium (out, fight) is not a subgame perfect equilibrium ¨  Strategy „fight“ is not optimal following

the history „in“ n  Nash equilibrium (in, acquiesce) is a

subgame perfect equilibrium


75 Subgame perfect equ. and Nash equ.

Every subgame perfect equilibrium is a Nash equilibrium

n  A subgame perfect equilibrium generates a Nash equilibrium in every subgame

n  In a Nash equilibrium every player’s strategy is optimal, given the other players’ strategies, in the whole game

n  but it may not be optimal in some subgames n  however, it is optimal in any subgame that is reached when the players follow

their strategies.


76 Backward induction

n  In a game with a finite horizon the set of subgame perfect equilibria may be found directly by using backward induction

n  Start by finding the optimal actions of the players who move in the subgames of length one

n  given those actions, we find the optimal actions of the players who move first in the subgames of length two, etc.

n  if in a subgame there are multiple actions that lead to the same payoff for that player, consider all those actions separately.

1

2

C D

E F

2,1 3,0 1,3 0,2

HG2


77 Backward induction - Exercises

1

2

C E

F G

3,0 1,0 1,3 2,2

KJ2

2,1 1,1

IH2

D

1

2C D

E F 2,0

3,1 0,0 1,2

H G 1

1,2

J I 1


78 Backward induction - Exercise

Exercise: n  2 players, 21 flags n  players can sequentially remove 1,2

or 3 flags n  the player who removes the last

flag wins

79

GAMES WITH PERFECT INFORMATION

EXTENSIVE FORM GAMES ILLUSTRATIONS


80 Product Choice Problem

n  What are the NE? n  Draw the game tree with firm 1 going first

n  what are the subgame perfect NE? n  change the order of action of the firms


81 Stackelberg’s model of duopoly

n  Stackelberg’s duopoly game (compare to Cournot model) n  one firm goes first, the other reacts based on the other’s quantity

¨  Assume inverse demand function:

¨  if firm 1 goes first, how much would they produce? ¨  compare to Cournot equilibrium ¨ would you want to go first or second?

¨  and cost functions:


82 Properties of subgame perfect equilibrium n  First-mover advantage

¨  for any cost and inverse demand functions for which firm 2 has a unique best response to each output of firm 1, firm 1 is at least as well off in any SPE of Stackelberg’s game as it is in any NE of Cournot’s game.

© Osborne 2004

¨  what if firm 1 could change its quantity again?

n  and if firm 2 knew this and would change also?

n  where does this lead? ¨  value of commitment


83 (Empty) Threats and Commitment

n  Incumbent (I) firm announces price war in case potential entrant (C) enters the market

n  Extensive form game looks as follows:

n  Strategic form game can be written as:

C

Iin out

acc fight

5,5 -1,-1

0,10

n  Is “fight” (in case C plays “in”) a credible threat?

n  What are Nash equilibria? n  Are they subgame perfect?


84 (Empty) Threats and Commitment

n  Can the incumbent make the threat credible? ¨  needs to change the payoffs in the game! ¨  invest in excess capacity, invest in consumer loyalty ¨  this decreases payoffs in case E stays out or enters without fight ¨  increases payoffs in case of fight

C

Iin out

acc fight

5,5-c -1,-1+d

0,10-c

n  I will fight if 5-c < -1+d n  if this holds, then (out, fight) is a SPNE

85

Markets With Asymmetric Information or The Economics of Information


86 Information

n  Until now we assumed complete information ¨  often some agents know more than others ¨  ASYMMETRIC INFORMATION

¨  Seller or producer knows more about the quality of a product than

the buyer ¨  Managers know more about costs, competitive position and

investment opportunities than firm owners

¨  asymmetric information explains many institutional arrangements in societies

n  guarantees, incentive schemes in contracts, etc.


87 Quality Uncertainty n  Example: Market for Lemons

¨  two kinds of cars: high quality and low quality ¨  sellers and buyers can tell the quality of a car ¨  two markets emerge

© Pearson Int.


88 Quality Uncertainty n  what if sellers know more about quality than buyers?

¨  buyers will view all cars as medium quality, i.e. think that there is a 50-50 chance that the car is of high quality

¨  perception different than expectation ¨  eventually high quality cars will be driven out of the market ¨  market failure

Medium quality cars sell for

$7500, selling 25,000 high quality and 75,000 low

quality.

© Pearson Int.


89 Implications of Asymmetric Information

ADVERSE SELECTION n  is a form of market failure resulting when products of different

qualities are sold at a single price because of asymmetric information ¨  low quality goods can drive high quality goods out of the market

(lemons problem) ¨  Hence, too many low and too few high quality goods are sold ¨  market fails to produce mutually beneficial trade

n  Applications ¨  Market for Insurance

n  insured person knows more about health status than insurance company ¨  Market for Credit

n  borrower knows more about his risk attitude than bank ¨  many other situations

n  retail stores, dealers of paintings, craftsmen (electricians), restaurants, …


90 Market Signaling

n  what can high-quality seller/agents do? n  MARKET SIGNALING

¨  is the process of one agent using signals to convey information to the other agent about the true state

¨  sellers use signals to inform buyers about product’s quality n  use guarantees and warranties n  cost of warranties to low-quality producers might be too high n  importance of reputation and standardization

¨  low-risk individuals try to inform insurers about their risk status ¨  workers try to signal employers their productivity

n  weak vs. strong signal

n  signals may be inaccurate ¨  design mechanisms to give incentives to let the agents reveal their

true situation ¨  separating equilibrium


91 Market Signaling – Example

n  2 types of workers ¨  MP1 = 1; MP2 = 2 ¨  Price of good: 10000 ¨  Expected employment time: 10 years ¨  AP1+2 = 1.5 ¨  wage 15000

n  if type was known each type would get its marginal product ¨  firm tries to distinguish workers on basis of education

n  y is the number of years educated n  no other attributes of workers used


92 Market Signaling – Example n  Education cost functions

¨  C1(y) = 40000y ¨  C2(y)= 20000y ¨  education does not increase a worker’s productivity

n  it is only a value as a signal ¨  What kind of equilibrium do you find?

© Pearson Int.


93 Moral Hazard n  other problems from asymmetric information, besides adverse

selection, ¨  what might happen if you buy a life-insurance? ¨  what might happen if you buy an insurance against theft? ¨  what might happen if you finally got the job?

n  MORAL HAZARD ¨  occurs when a party/agent whose actions are unobserved affects

the probability or magnitude of a payment ¨  alters the ability of markets to allocate resources efficiently

n  moral hazard not only changes behavior n  it also creates economic inefficiency


94 Moral Hazard n  Example

¨  insurance cost of driving a car is $0.50 per mile if observed by insurance company (plus $1 for fuel, taxes, etc.)

¨  if miles not observed, any additional mile does not incur the $0.50 cost but only the $1 cost for fuel, taxes, etc.

¨  will increase the quantity above efficient level

© Pearson Int.


95 The Principal-Agent Problem

n  Asymmetric information often applies to situations were one person (principal) hires another person (agent) to make economic decisions. ¨  Agency relationship exists whenever there is an arrangement in which

one person’s welfare depends on what another person does. ¨  Principals cannot monitor the productivity of agents perfectly

n  PRINCIPAL-AGENT-PROBLEM ¨  agents pursue their own goals, rather than the goals of the principal

n  owners hiring managers n  patients hiring doctors to decide on treatment n  investors hiring financial advisors n  car owner/buyer hiring mechanic

n  Problem of diverging incentives ¨  Principal needs mechanism to provide incentive for agent to work in

principal’s interest n  e.g. reward structure based on long-term performance


96 The Principal-Agent Problem n  Example

¨  managing a firm out of the principal’s vs. the agent’s view

© Nicholson


97 The Principal-Agent Problem n  How can owners design reward systems so that managers and

workers come as close as possible to meeting the owners’ goals? Example: ¨  manager can use low effort (a=0) or high effort (a=1) ¨  firm’s profit also depends on luck, however firm has incomplete

information

bad luck good luck low effort

(a=0) 10,000 20,000

high effort (a=1) 20,000 40,000

¨  manager’s cost function is c=10,000a and she wants to maximize wage ¨  owner’s goal is to maximize expected profit, given uncertainty and

inability to monitor the manager ¨  what is the optimal payment scheme?


98 Asymmetric Information in Labor Markets n  Competitive labor market would not allow for unemployment

¨  why do still most countries experience unemployment? ¨  efficiency wage theory

n  can explain presence of unemployment and wage discrimination n  labor productivity depends on the wage rate

¨  nutritional reasons in developing countries

n  In developed countries use shirking model ¨  perfectly competitive market - workers can work or shirk ¨  monitoring workers is costly or impossibility – imperfect information ¨  what if shirkers are detected?

n  wage at market clearing rate gives incentive to shirk - firm pays more to make loss from shirking higher

¨  wage at which no shirking occurs is the efficiency wage n  but what if all firms pay efficiency wages? n  is there again an incentive to shirk?


99 Asymmetric Information in Labor Markets ¨  Unemployment can arise in otherwise competitive labor markets when

employers cannot accurately monitor workers. ¨  “no shirking constraint” (NSC) gives the wage necessary to keep workers

from shirking. What is the optimal payment scheme? ¨  The firm hires Le workers (at a higher than competitive efficiency wage

we), creating L* − Le of unemployment.

© Pearson Int.


100 Asymmetric Information in Labor Markets

One of the early examples of the payment of efficiency wages can be found in the history of Ford Motor Company.

Ford needed to maintain a stable workforce, and Henry Ford (and his business partner James Couzens) provided it.

In 1914, when the going wage for a day’s work in industry averaged between $2 and $3, Ford introduced a pay policy of $5 a day. The policy was prompted by improved labor efficiency, not generosity.

Although Henry Ford was attacked for it, his policy succeeded. His workforce did become more stable, and the publicity helped Ford’s sales.

101

Externalities and Public Goods


102 Externalities n  You are having a party and your neighbor is upset because of the

loud music. What happens here? n  You are imposing an EXTERNALITY on your neighbor

¨  the effects of production and consumption activities not directly reflected in the market

¨  Negative externalities n  an action by one party imposes a cost on another party n  pollution of river by firm upstream affects firms downstream n  noise of airplanes affects citizens n  negative effect not taken into account by pollutant

¨  Positive externalities n  someone’s beautiful garden from which all can benefit (by looking at it)

¨  What if I buy some bread and this raises the price of bread for you? n  pecuniary externalities


103 Negative Externalities and Inefficiency n  Externalities create an external cost not represented in the firm’s

cost structure ¨  marginal external cost (MEC)

n  Marginal social cost (MSC) = MC + MEC

© Pearson Int.


104 Positive Externalities

© Pearson Int.


105 Externalities n  Mathematical Model of a production externality

¨  firm 1 produces output q selling it in a competitive market ¨  production of q imposes a cost e(q) on firm 2

equilibrium output q1 is given by P = c’(q1). n  firm 1 takes only private costs into account but not social costs n  what is the efficient amount of output?

¨  merge the two firms to internalize the externality

leads to a FOC in which price equals marginal social cost


106 Correcting Market Failure n  What can the state do to correct market failure? n  Model assumptions:

¨  Negative externality, pollution by firm ¨  output and emission decisions are independent

n  no link between quantity of emissions and quantity of output ¨  marginal external cost (MEC) of emissions is upward sloping

n  substantially increasing harm as pollution increases ¨  marginal cost of abating emissions (MCA) is downward sloping

n  if emissions are high, there is little cost to controlling them, but large reductions require costly changes in production process


107 Correcting Market Failure

© Pearson Int.


108 Correcting Market Failure n  Three ways to make firms reduce emissions to an efficient level

¨  Emission standards n  legal limit on quantity of emissions

¨  Emission fees n  charge levied on each unit of emissions

¨  Transferable emission permits n  permits to be traded on a market

© Pearson Int.



n  What is better? Standard or Fees? ¨  Assumptions

n  policymakers have asymmetric information n  same fee or standard required for all firms

© Pearson Int.



¨  optimal fee is $8 ¨  now incomplete information

n  what if 12,5% failure on standard (standard of 9)?

n  what if 12,5% failure on fee (fee of $7)?

¨  advantage of standards n  when MSC is steep and MCA flat and

limited information about efficient fee and efficient standard

7

9

© Pearson Int.


111 Correcting Market Failure n  Transferable Emissions Permits

¨  policymaker determines the level of emissions and number of permits ¨  permits are traded in a market ¨  help develop a competitive market for externalities

n  firms with high abatement costs will purchase permits from firms with low abatement costs

¨  combines both, standards and fees n  allows pollution abatement to be achieved at minimum cost

¨  Examples n  Australian carbon trading scheme n  Kyoto protocol


112 Emissions Trading - Example

Price of Tradeable Emissions Permits

n  Limiting Sulfur Dioxide emissions since 1990s n  Expected permit price: $300

¨  but initially much lower because less costly to reduce emissions (cheaper mining of low sulfur coal).

n  Sharp increase in 2005 ¨  as low sulfur coal price increased and more power plants were built leading to more permit

demand

© Pearson Int.


113 Recycling n  What is the efficient amount of recycling?

¨  households decide on whether to simply dispose certain waste (e.g. glass) or put it into a recycling container

n  simple disposal usually has low cost to households (MC) and seems to be constant with the amount of waste

¨  social costs of disposal (MSC) are much larger and increase with the amount of waste

¨  marginal costs of recycling (MCR) n  are increasing with the amount that needs to be reduced (right-left) as

collection, separation and cleaning costs grow at increasing rate

© Pearson Int.


114 Recycling n  What is the efficient amount of recycling?

¨  where MSC intersects MCR n  What is the market outcome?

¨  where MC intersects MCR ¨  hence the amount m1 is not socially optimal

n  How could we get to the social optimum? ¨  increase the private costs so that MC intersects MCR at the

socially optimal amount ¨  refundable deposits

© Pearson Int.


115 Stock externalities n  so far externalities from flows of harmful pollution n  sometimes damage not from emissions flow, but from the

accumulated stock of the pollutant. ¨  e.g. global warming, seen to result from the accumulation of

greenhouse gases (GHG) in the atmosphere n  does not cause severe immediate harm n  rather the stock of accumulated GHG ultimately causes harm n  stock will remain high even if current emission were reduced to zero

¨  also positive stock externalities possible n  e.g. the stock of knowledge resulting from investments in R&D

¨  need to compare the present discounted value (PDV) of additional profits resulting from any investment to the cost of the investment

n  i.e. need to calculate the net present value (NPV) n  cost-benefit analysis n  helps to decide whether an investment is economically justified


116 Stock externalities n  How does a stock built up?

¨  stock of pollutant, S ¨  present years emissions, E ¨  fraction of stock that dissipates, δ ¨  first year’s stock is first year’s emissions

¨  in second year, stock equals that year’s emissions plus non-dissipated stock from the first year

¨  in any year t

¨  for emissions at constant rate E, after N years the stock will be

¨  for N going to infinity, the stock approaches a long-run equilibrium


117 Stock externalities - Example

¨  100 units of pollutant emitted each year, i.e. Ei = 100 ¨  rate of stock dissipates by 2 percent per year, i.e. δ = 0.02 ¨  stock originally zero, i.e. S0 = 0 ¨  stock creates damage (health costs, etc.) equal to 1 mill. per unit ¨  annual cost of reducing emissions to zero is 15 mill. per unit

© Pearson Int.


118 Stock externalities - Example ¨  Does a policy of zero emission make (economic) sense? ¨  need to calculate the NPV (assuming social rate of discount R)

n  costs and benefits of a policy apply to society as a whole n  use opportunity cost to society of receiving an economic benefit in the

future rather than today n  little agreement on how to use it and its size

¨  NPV depends on discount rate, R, and dissipation rate, δ ¨  numbers for different combinations in the following table:

© Pearson Int.


119 Global warming n  Atmospheric concentration of GHG dramatically increased over

the past century ¨  even if current GHG emissions are stabilized, atmospheric

concentration continues to increase ¨  reduction of current GHG emissions

n  increases costs today with benefits in some 50 years or more n  economic predictions heavily depending on social rate of discount

© Pearson Int.


120 Externalities and Property Rights n  active government regulation not only way to deal with externalities n  alternative approach: PROPERTY RIGHTS

¨  legal rules stating what people or firms may do with their property ¨  Example: fishermen owning the river

n  fisherman can demand compensation from firm upstream polluting the river n  hence there is a cost to the firm upstream from polluting the river in form

of this compensation n  costs are internalized n  via bargaining economic efficiency can be achieved without government

intervention

n  COASE THEOREM ¨  if property rights are well specified and parties can bargain without

cost and to their mutual advantage, the resulting outcome will be efficient, regardless of how the property rights are specified

n  is bargaining really without cost and are property rights clearly specified?


121 Externalities and Property Rights

¨  what do you think will happen n  if firm has property right? n  if fishermen have property right? n  would they want to stick at those

levels? n  where do we find the socially

efficient level of emissions?

© Pearson Int.


122 Common Property Resources n  Sometimes externalities occur if resources can be used without

payment n  COMMON PROPERTY RESOURCE

¨  a resource to which anyone has free access ¨  air/water/land/fish/animal population/minerals/etc.

n  inefficiencies might occur n  prisoners’ dilemma

¨  Private cost of fishermen understates the true cost to the society n  fishermen fish until marginal private benefit is equal to marginal private

cost n  no fisherman takes into account how its own fishing affects others’

expectation of catching fish n  too many fish will be caught


123 Common Property Resources

n  Possible solution to install private (or government) ownership ¨  owner sets fee for use of resource equal to the MC of depleting the stock

¨  what is the private optimum? ¨  what is the social optimum?

© Pearson Int.


124 Public Goods

n  are there any reasons for government replacing private firms as producers?

n  PUBLIC GOODS ¨  Nonexclusive

n  people cannot be excluded from its consumption n  difficult or impossible to charge for its use

¨  Nonrival n  marginal cost of its provision to an additional consumer is zero n  beware of congestion

¨  pure public goods provide benefits to people at zero marginal cost and no one can be excluded from enjoying them.

¨  in contrast, a private good can be excluded and is rival!


125 Public Goods

n  where is the efficient level of provision of a private good? ¨  marginal benefit for a consumer = marginal cost of production

n  where is the efficient level of provision of a public good? ¨  marginal benefit for all consumers = marginal cost of production ¨  Samuelson condition

exclusive

Yes No

rival

Yes

No


126 Public Goods

¨  how much would they consume if this was a private good?

¨  what is the socially optimal level of provision?

¨  why does it seem difficult to privately provide a public good?

© Pearson Int.


127 Public Goods n  Private provision of public goods difficult n  free-riding

¨  free-rider is someone who does not pay for a nonexclusive good in the expectation that others will

¨  incentive problem – would you communicate your true benefit? ¨  government will take over the production of a public good and set

taxes and/or fees accordingly ¨  determining how much of a public good to provide – when free-riding

is possible – is difficult n  voting n  mechanism design


128 Private preferences for public goods ¨  efficient level of education determined by summing the willingness to

pay of the different individuals (net of tax payments) n  leads to an aggregate willingness to pay curve (AW) n  hence the efficient outcome is where AW is maximized, i.e. at a spending

of $1200 n  is this also the voting outcome? n  recall median voter theorem

© Pearson Int.

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