Basic Elements of Noncooperative Gameshrtdmrt2/Teaching/GT_2015_19/L2.pdf · Introduction...

Basic Elements of Noncooperative Games

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

May 28th, 2015

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Introduction

2 What is a Game?

3 The Extensive Form Representation of a Game

4 Strategies and the Normal Form Representation of a Game

5 Randomized Choices

6 Exercises

C. Hurtado (UIUC - Economics) Game Theory

Introduction

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises

C. Hurtado (UIUC - Economics) Game Theory 1 / 32

Introduction

Introduction

There are two leading frameworks for analyzing games: cooperative andnoncooperative.This course focuses on noncooperative game theory, which dominates applications.But even if not, you should be aware that cooperative game theory exists, and isbetter suited to analyzing some economic settings, e.g. where the structure of thegame is unclear or unobservable, and it is desired to make predictions that arerobust to it.Cooperative game theory assumes rationality, unlimited communication, andunlimited ability to make agreements.Its goal is to characterize the limits of the set of possible cooperative agreementsthat might emerge from rational bargaining.


Introduction

Introduction

Noncooperative game theory also assumes rationality.Noncooperative game theory replaces cooperative game theory’s assumptions ofunlimited communication and ability to make agreements with a fully detailedmodel of the situation and a detailed model of how rational players will behave init.Its goal is to use rationality, augmented by the ”rational expectations” notion ofNash equilibrium, to predict or explain outcomes from the data of the situation.As a result, noncooperative game theory is used for normative purposes in someapplications, such as mechanism design.Some applications of noncooperative game theory involve predicting which settingsare better for fostering cooperation.This is done by making behavioral assumptions at the individual level(”methodological individualism”), thereby requiring cooperation to emerge (if atall) as the outcome of explicitly modeled, independent decisions by individuals inresponse to explicitly modeled institutions.


Introduction

Introduction

In game theory, maintaining a clear distinction between the structure of a gameand behavioral assumptions about how players respond to it is analytically asimportant as keeping preferences conceptually separate from feasibility in decisiontheory.We will first develop a language to describe the structure of a noncooperativegame.We will then develop a language to describe assumptions about how playersbehave in games, gradually refining the notion of what it means to make a rationaldecision.In the process we will illustrate how game theory can elucidate questions ineconomics.As you learn to describe the structure, please bear in mind that the goal is to givethe analyst enough information about the game to formalize the idea of a rationaldecision. (This may help you be patient about not yet knowing exactly what itmeans to be rational.)


What is a Game?

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises


What is a Game?

What is a Game?

From the noncooperative point of view, a game is a multi-person decision situationdefined by its structure, which includes:

- Players: Independent decision makers- Rules: Which specify the order of players’ decisions, their feasible decisions

at each point they are called upon to make one, and the information theyhave at such points.

- Outcome: How players’ decisions jointly determine the physical outcome.- Preferences: players’ preferences over outcomes.

Assume that the numbers of players, feasible decisions, and time periods are finite.These can be relaxed, and it will be relaxed for decisions and time periods.Preferences over outcomes are modeled just as in decision theory.Preferences can be extended to handle shared uncertainty about how players’decisions determine the outcome as in decision theory, by assigning vonNeumann-Morgenstern utilities, or payoffs, to outcomes and assuming that playersmaximize expected payoff.


What is a Game?

What is a Game?

Assume for now that players face no uncertainty about the structure other thanshared uncertainty about how their decisions determine the outcome, that playersknow that no player faces any other uncertainty, that players know that they know,and so on; i.e. that the structure is common knowledge.It is essential that a player’s decisions be feasible independent of others’ decisions;e.g. ”wrestle with player 2” may be not a well-defined decision, although ”try towrestle with player 2” can be well-defined.It is essential that specifying all of each player’s decisions should completelydetermine an outcome (or at least a shared probability distribution over outcomes)in the game.If a specification of the structure of a game does not pass these tests, it must bemodified until it does.Example: If you object to a game analysis is that players are not really required toparticipate in the game, the (only!) remedy is to explicitly add a player’s decisionwhether to participate to the game, and then to insist that it be explained by thesame principles of behavior the analysis uses to explain players’ other decisions.


What is a Game?

Examples

Matching Pennies (version A).Players: There are two players, denoted 1 and 2.Rules: Each player simultaneously puts a penny down, either heads up or tails up.Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.Matching Pennies (version B).Players: There are two players, denoted 1 and 2.Rules: Player 1 puts a penny down, either heads up or tails up. Then, Player 2puts a penny down, either heads up or tails up.Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.


What is a Game?

Examples

Matching Pennies (version C).Players: There are two players, denoted 1 and 2.Rules: Player 1 puts a penny down, either heads up or tails up, without lettingplayer 2 know his decision. Player 2 puts a penny down, either heads up or tails up.Outcomes: If the two pennies match, player 1 pays 1 dollar to player 2; otherwise,player 2 pays 1 dollar to player 1.Matching Pennies (version D).Players: There are two players, denoted 1 and 2.Rules: Players flip a fair coin to decide who begins. The looser puts a penny down,either heads up or tails up. Then, the winner puts a penny down, either heads upor tails up.Outcomes: If the two pennies match, the looser pays 1 dollar to player 2;otherwise, the winner pays 1 dollar to player 1.


The Extensive Form Representation of a Game

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises




Some games that are important in economics have simultaneous moves.”Simultaneous” means strategically simultaneous, in the sense that players’decisions are made without knowledge of others’ decisions.It need not mean literal synchronicity, although that is sufficient for strategicsimultaneity.But many important games have at least some sequential decisions, with somelater decisions made with knowledge of others’ earlier decisions.We need a way to describe and analyze both kinds of game.One way to describe either kind of game is via the extensive form or game tree,which shows a game’s sequence of decisions, information, outcomes, and payoffs.




A version of Matching Pennies with sequential decisions, in which Player 1 movesfirst and player 2 observes 1’s decision before 2 chooses his decision.




We can represent the usual Matching Pennies with simultaneous decisions byintroducing an information set, which includes the decision nodes a player cannotdistinguish and at which he must therefore make the same decision, as in thecircled nodes.




The order in which simultaneous decision nodes are listed has some flexibility, as inprevious case, where player 2 could have been at the top.For sequential decisions the order must respect the timing of information flows.(Information about decisions already made, as opposed to predictions of futuredecisions, has no reverse gear.)All decision nodes in an information set must belong to the same player and havethe same set of feasible decisions. (Why?)Players are normally assumed necessarily to have perfect recall of their own pastdecisions (and other information). If so, the tree must reflect this.

DefinitionA game is one of perfect information if each information set contains a single decisionnode. Otherwhise, it is a game of imperfect information.



The Extensive Form Representation of a GameThis is an example of a game with simultaneous decision nodes and players withperfect recall of their own past decisions.




This is an example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.




This is another example of a game with simultaneous decision nodes and playerswithout perfect recall of their own past decisions.




Shared uncertainty (in economics ”symmetric information”) can be modeled byintroducing moves by an artificial player (without preferences) called Nature, whochooses the structure of the game randomly, with commonly known probabilities.


Strategies and the Normal Form Representation of a Game

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises




For sequential games it is important to distinguish strategies from decisions oractions.A strategy is a complete contingent plan for playing the game, which specifies afeasible decision for each of a player’s information sets in the game.Recall that his decision must be the same for each decision node in an informationset.A strategy is like a detailed manual of actions, not like a single decision or action.




It is assumed that that conditional on what a player observes, he can predict theprobability distributions of his own and others’ future decisions and theirconsequences.If players have this kind of foresight, then their rational sequential decision-makingin ”real time” should yield exactly the same distribution of decisions assimultaneous choice of fully contingent strategies at the start of play.The player writes his own manual of actions. Then he will give you (a neutralreferee) the manual and let you play out the game. You will tell him who won.Because strategies are complete contingent plans, players must be thought of aschoosing them simultaneously (without observing others’ strategies),independently, and irrevocably at the start of play.




Why a strategy must be a complete contingent plan, specifying decisions even fora player’s own nodes that he knows will be ruled out by his own earlier decisions?Otherwise, other players’ strategies would not contain enough information for aplayer to evaluate the consequences of his own alternative strategies.We would then be unable to correctly formalize the idea that a strategy choice isrational.Putting the point in an only seemingly different way, in individual decision theory,zero probability events can be ignored as irrelevant, at least for expected-utilitymaximizers.But in games zero-probability events cannot be ignored because what has zeroprobability is endogenously determined by players’ strategies.




Player 2 strategies:Strategy 1 (s1): Play H if player 1 plays H; Play H if player 1 plays TStrategy 2 (s2): Play H if player 1 plays H; Play T if player 1 plays TStrategy 3 (s3): Play T if player 1 plays H; Play H if player 1 plays TStrategy 4 (s4): Play T if player 1 plays H; Play T if player 1 plays T




A game maps strategy profiles (one for each player) into payoffs (with outcomesimplicit).A game form maps strategy profiles into outcomes, without specifying payoffs.Specifying strategies make it possible to describe an extensive-form game’srelationship between strategy profiles and payoffs by its (unique) normal form orpayoff matrix or (usually when strategies are continuously variable) payoff function.



Strategies and the Normal Form Representation of a GameThe mapping from the normal to the extensive form isn’t univalent: the normalform for Matching Pennies version B has possible extensive forms other than theone depicted before:


Randomized Choices

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises


Randomized Choices

Randomized Choices

In game theory it is useful to extend the idea of strategy from the unrandomized(pure) notion we have considered to allow mixed strategies (randomized strategychoices).Example: Matching Pennies Version C plainly has no appealing pure strategies,but there is a convincingly appealing way to play using mixed strategies:randomizing 50-50. (Why?)Our definitions apply to mixed as well as pure strategies, given that theuncertainty about outcomes that mixed strategies cause is handled (just as forother kinds of uncertainty) by assigning payoffs to outcomes so that rationalplayers maximize their expected payoffs.Mixed strategies will enable us to show that (reasonably well-behaved) gamesalways have rational strategy combinations.In extensive-form games with perfect recall, mixed strategies are equivalent tobehavior strategies, probability distributions over pure decisions at each node(Kuhn’s Theorem; see MWG problem 7.E.1).


Exercises

On the Agenda

1 Introduction

2 What is a Game?




6 Exercises


Exercises

Exercises

Exercise 1. In a game where player i has N information sets indexed n = 1, · · · , Nand Mn possible actions at information set n, how many strategies does player ihave?Exercise 2. Depict the normal formm of Matching Pennies Version C.


Exercises

Exercises

Exercise 3. Consider the followign two-player (excluding payoffs):

a) What are player 1’s possible strategies? player 2’s?b) Suppose that we change the game by merging the information set of player 1’s

second round of moves (so that all the four nodes are now in a single informationset). Argue why the game is no longer one of perfect recall.


Basic Elements of Noncooperative Gameshrtdmrt2/Teaching/GT_2015_19/L2.pdf · Introduction...

Documents

Transcript of Basic Elements of Noncooperative Gameshrtdmrt2/Teaching/GT_2015_19/L2.pdf · Introduction...