Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost.

Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost essential Game Theory: Dynamic Prerequisites

Frank Cowell: Microeconomics Overview... Basic structure Equilibrium issues Applications Repeated Games Embedding the game in context…

Frank Cowell: Microeconomics Introduction Another examination of the role of time Another examination of the role of time Dynamic analysis can be difficult Dynamic analysis can be difficult  more than a few stages…  …can lead to complicated analysis of equilibrium We need an alternative approach We need an alternative approach  but one that preserves basic insights of dynamic games  such as subgame-perfect equilibrium Build on the idea of dynamic games Build on the idea of dynamic games  introduce a jump  from the case of comparatively few stages…  …to the case of arbitrarily many

Frank Cowell: Microeconomics Repeated games The alternative approach The alternative approach  take a series of the same game  embed it within a time-line structure Basic idea is simple Basic idea is simple  by connecting multiple instances of an atemporal game…  …model a repeated encounter between the players in the same situation of economic conflict Raises important questions Raises important questions  how does this structure differ from an atemporal model?  how does the repetition of a game differ from a single play?  how does it differ from a collection of unrelated games of identical structure with identical players?

Frank Cowell: Microeconomics History Why is the time-line different from a collection of unrelated games? Why is the time-line different from a collection of unrelated games? The key is history The key is history  history at any point on the timeline…  …is the information about actual play…  …accumulated up to that point History can affect the nature of the game History can affect the nature of the game  at any stage all players can know all the accumulated information  strategies can be conditioned on this information History can play a role in the equilibrium History can play a role in the equilibrium  some outcomes that aren’t equilibria in a single encounter…  …may yet be equilibria outcomes in the repeated game  the game’s history is used to support such outcomes

Frank Cowell: Microeconomics Repeated games: Structure The stage game The stage game  take an instant in time  specify a simultaneous-move game  payoffs completely specified by actions within the game Repeat the stage game indefinitely Repeat the stage game indefinitely  there’s an instance of the stage game at time 0,1,2,…,t,…  the possible payoffs are also repeated for each t  payoffs at t depends on actions in stage game at t A modified strategic environment A modified strategic environment  all previous actions assumed as common knowledge  so agents’ strategies can be conditioned on this information Modifies equilibrium behaviour and outcome? Modifies equilibrium behaviour and outcome?

Frank Cowell: Microeconomics Equilibrium Simplified structure has potential advantages Simplified structure has potential advantages  whether significant depends on nature of stage game  concern nature of equilibrium Possibilities for equilibrium Possibilities for equilibrium  new strategy combinations supportable as equilibria?  long-term cooperative outcomes…  …absent from a myopic analysis of a simple game Refinements of subgame perfection simplify the analysis: Refinements of subgame perfection simplify the analysis:  can rule out empty threats…  …and incredible promises  disregard irrelevant “might-have-beens”

Frank Cowell: Microeconomics Overview... Basic structure Equilibrium issues Applications Repeated Games Developing the basic concepts…

Frank Cowell: Microeconomics Equilibrium: an approach Focus on key question in repeated games: Focus on key question in repeated games:  how can rational players use the information from history?  need to address this to characterise equilibrium Illustrate a method in an argument by example Illustrate a method in an argument by example  Outline for the Prisoner's Dilemma game  same players face same outcomes from their actions that they may choose in periods 1, 2,..., t,.... Prisoner's Dilemma particularly instructive given: Prisoner's Dilemma particularly instructive given:  its importance in microeconomics  pessimistic outcome of an isolated round of the game

Frank Cowell: Microeconomics [ RIGHT ] 1,13,0 0,32,2[LEFT]Alf Bill [left] [right] Prisoner’s dilemma: Reminder   Payoffs in stage game   If Alf plays [RIGHT] then Bill’s best response is [right].   If Bill plays [right] then Alf’s best response is [RIGHT].   Nash Equilibrium   Outcome that Pareto dominates NE   The highlighted NE is inefficient   Could the Pareto-efficient outcome be an equilibrium in the repeated game?   Look at the structure…

Frank Cowell: Microeconomics Repeated Prisoner's dilemma Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) 2 2 1 1   Stage game (t=1)   Stage game (t=2) follows here…   or here… Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) 2 2   or here… Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) 2 2   or here… Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) 2 2   Repeat this structure indefinitely…?

Frank Cowell: Microeconomics Repeated Prisoner's dilemma......... Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1)......... Bill Alf [LEFT] [RIGHT] [left][right] [left][right] (2,2)(0,3)(3,0) (1,1) t t 1 1   The stage game…   …repeated though time Let's look at the detail

Frank Cowell: Microeconomics Repeated PD: payoffs To represent possibilities in long run: To represent possibilities in long run:  first consider payoffs available in the stage game  then those available through mixtures In the one-shot game payoffs simply represented In the one-shot game payoffs simply represented  it was enough to denote them as 0,…,3  purely ordinal…  …arbitrary monotonic changes of the payoffs have no effect Now we need a generalised notation Now we need a generalised notation  cardinal values of utility matter  we need to sum utilities, compare utility differences Evaluation of a payoff stream: Evaluation of a payoff stream:  suppose payoff to agent h in period t is  h (t)  value of (  h (1),  h (2),...,  h (t)...) is given by ∞ [1  ] ∑  t  1  h (t) t=1 t=1  where  is a discount factor 0 <  < 1

Frank Cowell: Microeconomics PD: stage game A generalised notation for the stage game A generalised notation for the stage game  consider actions and payoffs…  …in each of four fundamental cases Both socially irresponsible: Both socially irresponsible:  they play [RIGHT], [right]  get 0, 0  get   a  b  where  a > 0,  b > 0 Both socially responsible: Both socially responsible:  they play [LEFT],[left]  get ( *a, *b ) where *a >, *b >  get (  *a,  *b ) where  *a >  a,  *b >  b Only Alf socially responsible: Only Alf socially responsible:  they play [LEFT], [right]  get where > *b  get  b  where  b >  *b Only Bill socially responsible: Only Bill socially responsible:  they play [RIGHT], [left]  get where > *a  get  a  where  a >  *a A diagrammatic view

Frank Cowell: Microeconomics Repeated Prisoner’s dilemma payoffs U*U* aa bb 0   a  b     *a  *b  b b  a a    Space of utility payoffs   Payoffs for Prisoner's Dilemma   Nash-Equilibrium payoffs   Payoffs available through mixing   Feasible, superior points   "Efficient" outcomes   Payoffs Pareto-superior to NE

Frank Cowell: Microeconomics Choosing a strategy: setting Long-run advantage in the Pareto-efficient outcome Long-run advantage in the Pareto-efficient outcome  payoffs ( *a, *b ) in each period…  payoffs (  *a,  *b ) in each period…  …clearly better than in each period  …clearly better than   a  b  in each period Suppose the agents recognise the advantage Suppose the agents recognise the advantage  what actions would guarantee them this?  clearly they need to play [LEFT], [left] every period The problem is lack of trust: The problem is lack of trust:  they cannot trust each other…  …nor indeed themselves:  Alf tempted to be antisocial and get payoff  a by playing [RIGHT]  Alf tempted to be antisocial and get payoff  a by playing [RIGHT]  Bill has a similar temptation

Frank Cowell: Microeconomics Choosing a strategy: formulation Will a dominated outcome still be inevitable? Will a dominated outcome still be inevitable? Suppose each player adopts a strategy that Suppose each player adopts a strategy that 1. rewards the other party's responsible behaviour by responding with the action [left] 2. punishes antisocial behaviour with the action [right], thus generating the minimax payoffs 2. punishes antisocial behaviour with the action [right], thus generating the minimax payoffs   a   b  Known as a trigger strategy Known as a trigger strategy Why the strategy is powerful Why the strategy is powerful  punishment applies to every period after the one where the antisocial action occurred  if punishment invoked offender is “minimaxed for ever” Look at it in detail Look at it in detail

Frank Cowell: Microeconomics [RIGHT] Anything else Bill’s action in 0,…,t Alf Alf’s action at t+1 Repeated PD: trigger strategies   Take situation at t   First type of history   Response of other player to continue this history   Second type of history   Punishment response [LEFT] [left][left],…,[left] [right] Anything else Alf’s action in 0,…,t Bill Bill’s action at t+1 [left] [LEFT][LEFT],…,[LEFT] Will it work? sTasTasTasTa sTbsTbsTbsTb   Trigger strategies [s T a, s T b ]

Frank Cowell: Microeconomics Will the trigger strategy “work”? Utility gain from “misbehaving” at t:  a  −  *a Utility gain from “misbehaving” at t:  a  −  *a What is value at t of punishment from t+1 onwards? What is value at t of punishment from t+1 onwards?  Difference in utility per period:  *a  −  a  Discounted value of this in period t+1: V := [  *a  −  a ]/[1  −  ]  Value of this in period t:  V =  [  *a  −  a ]/[1  −  ] So agent chooses not to misbehave if So agent chooses not to misbehave if   a  −  *a  ≤  [  *a  −  a ]/[1  −  ] But this is only going to work for specific parameters But this is only going to work for specific parameters  value of  …  … relative to  a  a and  *a What values of discount factor will allow an equilibrium? What values of discount factor will allow an equilibrium?

Frank Cowell: Microeconomics Discounting and equilibrium For an equilibrium condition must be satisfied for both a and b For an equilibrium condition must be satisfied for both a and b Consider the situation of a Consider the situation of a Rearranging the condition from the previous slide: Rearranging the condition from the previous slide:   [  *a  −  a ] ≥ [1  −  [  a  −  *a  ]   [  a  −  a ] ≥ [  a  −  *a  ] Simplifying this the condition must be Simplifying this the condition must be   ≥  *a  where  a := [  a  −  *a  ] / [  a  −  a ] A similar result must also apply to agent b A similar result must also apply to agent b Therefore we must have the condition: Therefore we must have the condition:   ≥   where  := max {  a,  b }

Frank Cowell: Microeconomics Repeated PD: SPNE Assuming  ≥ , take the strategies [s T a, s T b ] prescribed by the Table Assuming  ≥ , take the strategies [s T a, s T b ] prescribed by the Table If there were antisocial behaviour at t consider the subgame that would then start at t+1 If there were antisocial behaviour at t consider the subgame that would then start at t+1  Alf could not increase his payoff by switching from [RIGHT] to [LEFT], given that Bill is playing [left]  a similar remark applies to Bill  so strategies imply a NE for this subgame  likewise for any subgame starting after t+1. But if [LEFT],[left] has been played in every period up till t: But if [LEFT],[left] has been played in every period up till t:  Alf would not wish to switch to [RIGHT]  a similar remark applies to Bill  again we have a NE So, if  is large enough, [s T a, s T b ] is a Subgame-Perfect Equilibrium So, if  is large enough, [s T a, s T b ] is a Subgame-Perfect Equilibrium  yields the payoffs (  *a, *b ) in every period  yields the payoffs (  *a,  *b ) in every period

Frank Cowell: Microeconomics Folk Theorem The outcome of the repeated PD is instructive The outcome of the repeated PD is instructive  illustrates an important result  …the Folk Theorem Strictly speaking a class of results Strictly speaking a class of results  finite/infinite games  different types of equilibrium concepts A standard version of the Theorem: A standard version of the Theorem:  In a two-person infinitely repeated game:  if the discount factor is sufficiently close to 1  any combination of actions observed in any finite number of stages…  …is the outcome of a subgame-perfect equilibrium

Frank Cowell: Microeconomics Assessment The Folk Theorem central to repeated games The Folk Theorem central to repeated games  perhaps better described as Folk Theorems  a class of results Clearly has considerable attraction Clearly has considerable attraction Put its significance in context Put its significance in context  makes relatively modest claims  gives a possibility result Only seen one example of the Folk Theorem Only seen one example of the Folk Theorem  let’s apply it…  …to well known oligopoly examples

Frank Cowell: Microeconomics Overview... Basic structure Equilibrium issues Applications Repeated Games Some well-known examples…

Frank Cowell: Microeconomics Cournot competition: repeated Start by reinterpreting PD as Cournot duopoly Start by reinterpreting PD as Cournot duopoly  two identical firms  firms can each choose one of two levels of output – [high] or [low]  can firms sustain a low-output (i.e. high-profit) equilibrium? Possible actions and outcomes in the stage game: Possible actions and outcomes in the stage game:  [HIGH], [high]: both firms get Cournot-Nash payoff  C   [LOW], [low]: both firms get joint-profit maximising payoff  J   C  [HIGH], [low]: payoffs are (  0) where   J Folk theorem: get SPNE with payoffs (  J,  J ) if  is large enough Folk theorem: get SPNE with payoffs (  J,  J ) if  is large enough  Critical value for the discount factor  is  −  J  −  J  =──────  −  C  −  C But we should say more But we should say more  Let’s review the standard Cournot diagram

Frank Cowell: Microeconomics q1q1 q2q2 2(·)2(·) 1(·)1(·) ll ll ll ll 0 (q C, q C ) 1 2 (q J, q J ) 1 2 Cournot stage game   Firm 2’s Iso-profit curves   Firm 2’s reaction function   Cournot-Nash equilibrium   Firm 1’s Iso-profit curves   Firm 1’s reaction function   Outputs with higher profits for both firms   Joint profit-maximising solution q2q2   Output that forces other firm’s profit to 0 ll ll ll ll q1q1

Frank Cowell: Microeconomics Repeated Cournot game: Punishment Standard Cournot model is richer than simple PD: Standard Cournot model is richer than simple PD:  action space for PD stage game just has the two output levels  continuum of output levels introduces further possibilities Minimax profit level for firm 1 in a Cournot duopoly Minimax profit level for firm 1 in a Cournot duopoly  is zero, not the NE outcome  C  arises where firm 2 sets output to  q 2 such that 1 makes no profit Imagine a deviation by firm 1 at time t Imagine a deviation by firm 1 at time t  raises q 1 above joint profit-max level Would minimax be used as punishment from t+1 to ∞? Would minimax be used as punishment from t+1 to ∞?  clearly (0,  q 2 ) is not on firm 2's reaction function  so cannot be best response by firm 2 to an action by firm 1  so it cannot belong to the NE of the subgame  everlasting minimax punishment is not credible in this case

Frank Cowell: Microeconomics Repeated Cournot game: Payoffs 11 22 0  C  C        Space of profits for the two firms   Cournot-Nash outcome   Joint-profit maximisation  J  J    Minimax outcomes   Payoffs available in repeated game Now review Bertrand compewtition

Frank Cowell: Microeconomics p2p2 c c pMpM pMpM ll ll p1p1 ll ll ll ll ll ll ll ll Bertrand stage game   Firm 1’s reaction function   Firm 2’s reaction function   Marginal cost pricing   Monopoly pricing   Nash equilibrium

Frank Cowell: Microeconomics Bertrand competition: repeated NE of the stage game: NE of the stage game:  set price equal to marginal cost c  results in zero profits NE outcome is the minimax outcome NE outcome is the minimax outcome  minimax outcome is implementable as a Nash equilibrium…  … in all the subgames following a defection from cooperation In repeated Bertrand competition In repeated Bertrand competition  firms set p M if acting “cooperatively”  split profits between them  if one firm deviates from this…  …others then set price to c Repeated Bertrand: result Repeated Bertrand: result  can enforce joint profit maximisation through trigger strategy…  …provided discount factor is large enough

Frank Cowell: Microeconomics Repeated Bertrand game: Payoffs 11 22 0 MM MM   Space of profits for the two firms   Bertrand-Nash outcome   Firm 1 as a monopoly   Firm 2 as a monopoly   Payoffs available in repeated game

Frank Cowell: Microeconomics Repeated games: summary New concepts: New concepts:  Stage game  History  The Folk Theorem  Trigger strategy What next? What next?  Games under uncertainty

Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost.

Similar presentations

Presentation on theme: "Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost."— Presentation transcript:

Similar presentations

About project

Feedback

Log in

Auth with social network:

Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost.

Similar presentations

Presentation on theme: "Frank Cowell: Microeconomics Repeated Games MICROECONOMICS Principles and Analysis Frank Cowell January 2007 Almost essential Game Theory: Dynamic Almost."— Presentation transcript:

Similar presentations

About project

Feedback