An Introduction to Game Theory and Basic Elements of ...hrtdmrt2/Teaching/GT_2016_19/L1.pdfAn...

An Introduction to Game Theory andBasic Elements of Noncooperative Games

Carlos Hurtado

Department of EconomicsUniversity of Illinois at Urbana-Champaign

[email protected]

Junel 7th, 2016

C. Hurtado (UIUC - Economics) Game Theory

mailto:[email protected]

On the Agenda

1 Introduction

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

5 Noncooperative Games

6 What is a Game?


Introduction

On the Agenda

1 Introduction





6 What is a Game?


Introduction

What do we do in Economic?

I What makes a theoretical model ”economics” is that the concepts we areanalyzing are taken from real life.

I Through the investigation of these concepts, we indeed try to understand realitybetter, and the models provide a language that enables us to think abouteconomic interactions in a systematic way.

I We do not view economic models as an attempt to describe exactly the world, orto provide tools for predicting the future.

I Although we will be studying formal concepts and models, they will always begiven an interpretation. An economic model differs substantially from a purelymathematical model in that it is a combination of a mathematical model and itsinterpretation.

I The word ”model” sounds more scientific than ”fable” or ”fairy tale”, but there isnot much difference between them. The author of a fable draws a parallel to asituation in real life and has some moral he wishes to impart to the reader.

C. Hurtado (UIUC - Economics) Game Theory 1 / 26

Introduction

What do we do in Economic?

I The fable is an imaginary situation that is somewhere between fantasy and reality.Any fable can be dismissed as being unrealistic or simplistic, but this is also thefable’s advantage.

I Being something between fantasy and reality, a fable is free of extraneous detailsand annoying diversions. In this unencumbered state, we can clearly discern whatcannot always be seen from the real world.

I On our return to reality, we are in possession of some sound advice or a relevantargument that can be used in the real world. We do exactly the same thing ineconomic theory.

I Thus, a good model in economic theory, like a good fable, identifies a number ofthemes and elucidates them. We perform thought exercises that are only looselyconnected to reality and have been stripped of most of their real-life characteristics.

I However, in a good model, as in a good fable, something significant remains.


Introduction

As if Rationality

I Rationality forms the basis of decision-making in the neoclassical school.

I Decision-makers are optimizers, given the constraints they find themselves in.

I Rationality assumes that decision-makers maximize things that give themhappiness and minimize things that give them pain.

I Implications:

- Narrows down the set of possible outcomes.

- Rational man is a clever individual.

- Rationality helps to predict the outcome of an economic system.

- Once economic agents have optimized their utility and reached a situationwhere they do not want deviate, the economic system reaches a stableoutcome: ’equilibrium’.


Introduction

Perfect Competition is a Benchmark

I Assumes the existence of many buyers and many sellers.

I Decision making is made independently and individually.

- Decisions are not made in coalitions (together/jointly)

- Decisions are not inter-dependent: Decision of one agent is neitherinfluenced by another agent, nor does it influence that of another agent.

I Independent and individual decision-making under perfect competition implies eachdecision-maker tries to do the best they can irrespective of what otherdecision-makers are doing. (really?)

I Perfect competition is a theoretical extreme. Like the ideal human bodytemperature of 98.4 degrees Fahrenheit it almost never exists. It is used as abenchmark to explain deviations from this ’perfect’ world.


What is Game Theory?

On the Agenda

1 Introduction





6 What is a Game?




I Branch of applied mathematics and economics that studies strategic situationswhere there are several players, with different goals, whose actions can affect oneanother.

I A game is any situation where multiple players can affect the outcome, a player isa stakeholder, a move or option is an action a player can take and, at the end ofthe game, the payoff for each player is the outcome.

I The value of game theory lies in understanding the interactions and likelyoutcomes when the end result is dependent on the actions of others who havepotentially conflicting motives.

I Game theory is mainly used in economics, political science, and psychology, as wellas logic, computer science, and biology.

I The subject first addressed zero-sum games, such that one person’s gains exactlyequal net losses of the other participant or participants. Today, game theoryapplies to a wide range of behavioral relations: the study of decision science,including both humans and non-humans.


A Brief History of Game Theory

On the Agenda

1 Introduction





6 What is a Game?




1713In a letter dated 13 November 1713 Francis Waldegrave provided the first known,minimax mixed strategy solution to a two-person game. Waldegrave wrote theletter, about a two-person version of a card game, to Pierre-Remond de Montmortwho in turn wrote to Nicolas Bernoulli, including in his letter a discussion of theWaldegrave solution.

1838The French economist Antoine Augustine Cournot discussed a duopoly where thetwo duopolists set their output based on residual demand.

1871In the first edition of his book The Descent of Man, and Selection in Relation toSex Charles Darwin gives the first (implicitly) game theoretic argument inevolutionary biology: If births of females are less common than males, females canexpect to have more offspring. Thus parents genetically disposed to producefemales tend to have more than the average numbers of grandchildren, hence,female births become more common. As the 1:1 sex ratio is approached, theadvantage associated with producing females dies away.




Intellectual debate between the French Mathematician Emile Borel (1871-1956)and the Hungarian Mathematician John Von Neumann (1903-1957):

Do zero-sum games have a solution?

1921Emile Borel published four notes on strategic games and an erratum to one ofthem. Borel gave the first modern formulation of a mixed strategy along withfinding the minimax solution for two-person games with three or five possiblestrategies. Initially he maintained that games with more possible strategies wouldnot have minimax solutions, but by 1927, he considered this an open question ashe had been unable to find a counterexample.

1928John von Neumann proved the minimax theorem. It states that every two-personzero-sum game with finitely many pure strategies for each player is determined, ie:when mixed strategies are admitted, this variety of game has precisely oneindividually rational payoff vector.




1934R.A. Fisher independently discovers Waldegrave’s solution to the card game. Fisherreported his work in the paper Randomisation and an Old Enigma of Card Play.

Intellectual debate between Hungarian Mathematician John Von Neumann(1903-1957) and he Austrian economist Oskar Morgenstern (1902-1977):

Can utility be quantified?

1944Theory of Games and Economic Behavior by John von Neumann and OskarMorgenstern is published. As well as expounding two-person zero sum theory thisbook is the seminal work in areas of game theory such as the notion of acooperative game, with transferable utility (TU), its coalitional form and its vonNeumann-Morgenstern stable sets. It was also the account of axiomatic utilitytheory given here that led to its wide spread adoption within economics.




Von Neumann and Morgenstern formally laid the foundations of Game Theory as abranch of applied mathematics. However, their effort failed to generate stir for tworeasons.

1. The Theory of Games and Economic Behavior was based on the notion ofzero-sum games. These are known as ’Games of Conflict’ or ’Non-CooperativeGames’. Most games in social sciences are non-zero-sum games.

2. It did not establish how equilibrium in games of interdependent decision-makingwould arise. The world did not have to wait too long for this solution.




1950Contributions to the Theory of Games I, H. W. Kuhn and A. W. Tucker eds.,published.

1950In January 1950 Melvin Dresher and Merrill Flood identified a game where it is inthe best interest for players to cooperate, but individual self-interest invokes themto not cooperate. The game reaches a bad equilibrium that is inferior to a superioroutcome that could have been reached and was available. This phenomenon wascanonized by Albert Tucker.

In the summer of 1950 Tucker was at Stanford University. He was working on aproblem in his room when a graduate student of psychology knocked and askedwhat he was doing. The answer was short: game theory. ”Why don’t you explainto us in a seminar”? Tucker used his now famous example of two thieves who wereput into separate cells and asked the same question by the judge. Tuckerchristened the phenomenon as The Prisoners’ Dilemma.



A Brief History of Game Theory1953

Extensive form games allow the modeler to specify the exact order in which playershave to make their decisions and to formulate the assumptions about theinformation possessed by the players in all stages of the game. In two papers,Extensive Games (1950) and Extensive Games and the Problem of Information(1953), H. W. Kuhn included the formulation of extensive form games which iscurrently used, and also some basic theorems pertaining to this class of games.

1953In four papers between 1950 and 1953 John Nash (1928 - 2015) made seminalcontributions to both non-cooperative game theory and to bargaining theory.

Note: Nash arrived at Princeton in the Fall of 1948 to start a PhD. He came toPrinceton with one of the shortest reference letters. His professor Richard Duffin(1909-1996) wrote just one sentence: ”This man is a genius.”Just eighteen months later, Nash submitted a 28 page doctoral dissertation. Thenumber 28 went on to become a superstitious number at Princeton. VonNeumann solved the mini-max theorem in 1928. Nash was born in 1928. Nash’sdoctoral dissertation was 28 pages.




1953In two papers, Equilibrium Points in N-Person Games (1950) and Non-cooperativeGames (1951), Nash proved the existence of a strategic equilibrium fornon-cooperative games -the Nash equilibrium- and proposed the ”Nash program”,in which he suggested approaching the study of cooperative games via theirreduction to non-cooperative form.

I This was the missing link to Von Neumann and Morgenstern’s magnum opus!!I These two papers established equilibrium where all players calculate their best

strategy based on what they assume is the best strategy of the other players.I Every finite game must have at least one solution such that once reached, no

player within the game will have an incentive to deviate from their chosen actions,given the actions of the other players in the game.

I The intuition behind Nash’s equilibrium is very simple: If all rational economicagents in a system are trying to do the best they can, assuming the others aredoing the same, the economic system must be in equilibrium such that no singleagent will want to unilaterally deviate from their position.

I A Nash equilibrium does not necessarily imply that a game will reach a solutionthat is the best possible solution for all players in the game.




1951George W. Brown described and discussed a simple iterative method forapproximating solutions of discrete zero-sum games in his paper Iterative Solutionsof Games by Fictitious Play.

1953The notion of the Core as a general solution concept was developed by L. S.Shapley (Rand Corporation research memorandum, Notes on the N-Person GameIII: Some Variants of the von-Neumann-Morgenstern Definition of Solution, RM-817, 1952) and D. B. Gillies (Some Theorems on N-Person Games, Ph.D. thesis,Department of Mathematics, Princeton University, 1953). The core is the set ofallocations that cannot be improved upon by any coalition.

50’sNear the end of this decade came the first studies of repeated games. The mainresult to appear at this time was the Folk Theorem. This states that theequilibrium outcomes in an infinitely repeated game coincide with the feasible andstrongly individually rational outcomes of the one-shot game on which it is based.Authorship of the theorem is obscure.


The Theory of Rational Choice

On the Agenda

1 Introduction





6 What is a Game?




I This theory is that a decision-maker chooses the best action according to herpreferences, among all the actions available to her.

I Her ”rationality” lies in the consistency of her decisions when faced with differentsets of available actions, not in the nature of her likes and dislikes.

Actions:- A set A consisting of all the actions that, under some circumstances, are

available to the decision-maker.- In any given situation the decision-maker is faced with a subset of A, from

which she must choose a single element.- The decision-maker knows this subset of available choices, and takes it as

given; in particular, the subset is not influenced by the decision-maker’spreferences.

- The set A could, for example, be the set of bundles of goods that thedecision-maker can possibly consume; given her income at any time, she isrestricted to choose from the subset of A containing the bundles she canafford.




Preferences and payoff functions:

- We assume that the decision-maker, when presented with any pair of actions,knows which of the pair she prefers, or knows that she regards both actionsas equally desirable (is ”indifferent between the actions”).

- Rational Choice (Transitivity): We assume further that these preferences areconsistent in the sense that if the decision-maker prefers the action a to theaction b, and the action b to the action c, then she prefers the action a tothe action c.

- Note that we do not rule out the possibility that a person’s preferences arealtruistic in the sense that how much she likes an outcome depends on someother person’s welfare.

- We can ”represent” the preferences by a payoff function, which associates anumber with each action in such a way that actions with higher numbers arepreferred.

a ≺ b ⇐⇒ u(a) < u(b)




A Digression on Transitivity:

- Aggregation of considerations as a source of intransitivity.

a �1 b �1 c;b �2 c �2 a; c �3 a �3 b ⇒ a � b � c � a

- The use of similarities as an obstacle to transitivity.

An individual may express indifference in a comparison between two elementsthat are too ”close” to be distinguishable.

I The theory of rational choice is that in any given situation the decision-makerchooses the member of the available subset of A that is best according to herpreferences.

I The action chosen by a decision-maker is at least as good, according to herpreferences, as every other available action.




I Exercise 1. (Altruistic preferences) Person 1 cares both about her income andabout person 2’s income. Precisely, the value she attaches to each unit of her ownincome is the same as the value she attaches to any two units of person 2’sincome. How do her preferences order the outcomes (1, 4), (2, 1), and (3, 0),where the first component in each case is person 1’s income and the secondcomponent is person 2’s income? Give a payoff function consistent with thesepreferences. Is that payoff unique?

I Exercise 2. (Alternative representations of preferences) Adecision-maker’spreferences over the set A = a, b, c are represented by the payoff function uforwhich u(a) = 0, u(b) = 1, and u(c) = 4. Are they also represented by thefunction v for which v(a) = -1, v(b) = 0, and v(c) = 2? How about the functionw for which w(a) = w(b) = 0 and w(c) = 8?




I Exercise 3. Deffine x ∼ y ⇐⇒ x � y and y � x . Denote the set of acctions byX . Define I(x) to be the set of all y in X for which y ∼ x . Show that the set (ofsets!) {I(x)|x ∈ X} is a partition of X . That is:

1. For all x and y , either I(x) = I(y) or I(x) ∩ I(y) = ∅.

2. For every x ∈ X , there is y ∈ X such that x ∈ I(y).




I Exercise 3 (Ans.). Deffine x ∼ y ⇐⇒ x � y and y � x .

- That is: x ∼ y ⇐⇒ ¬x � y or ¬y � x

- Let I(x) = {y ∈ X |x ∼ y}.

2. For every x ∈ X , it is clear that x ∼ x . Then, for every x ∈ X , there is atleast y = x ∈ X such that x ∈ I(y).

1. Let x , y ∈ X and assume that I(x) 6= I(y) and I(x) ∩ I(y) 6= ∅

There exist a z ∈ X , such that z ∈ I(x) and z ∈ I(z). That is, x ∼ z andy ∼ z.

Hence, x ∼ z ∼ y .

Therefore, I(x) = I(y) and I(x) 6= I(y) (Contradiction)


Noncooperative Games

On the Agenda

1 Introduction





6 What is a Game?




I There are two leading frameworks for analyzing games: cooperative andnoncooperative.

I This course focuses on noncooperative game theory, which dominates applications.

I 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.

I Cooperative game theory assumes rationality, unlimited communication, andunlimited ability to make agreements.

I Its goal is to characterize the limits of the set of possible cooperative agreementsthat might emerge from rational bargaining.




I Noncooperative game theory also assumes rationality.

I 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.

I 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.

I As a result, noncooperative game theory is used for normative purposes in someapplications, such as mechanism design.

I Some applications of noncooperative game theory involve predicting which settingsare better for fostering cooperation.

I 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.




I 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.

I We will first develop a language to describe the structure of a noncooperativegame.

I 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.

I In the process we will illustrate how game theory can elucidate questions ineconomics.

I 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





6 What is a Game?


What is a Game?

What is a Game?

I 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.

I Assume that the numbers of players, feasible decisions, and time periods are finite.

I These can be relaxed, and it will be relaxed for decisions and time periods.

I Preferences over outcomes are modeled just as in decision theory.

I 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?

I 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.

I 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.

I 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.

I If a specification of the structure of a game does not pass these tests, it must bemodified until it does.

I 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

I 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.

I Matching Pennies (version B).


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.



What is a Game?

Examples

I Matching Pennies (version C).


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.


I Matching Pennies (version D).


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.


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