Introduction to Game Theory

Home , Cooperative game theory

Carlos Hurtado

Department of Economics University of Illinois at Urbana-Champaign [email protected]

May 27th, 2015

C. Hurtado (UIUC - Economics) Game Theory On the Agenda

1 What do we do in Economic?

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

C. Hurtado (UIUC - Economics) Game Theory What do we do in Economic? On the Agenda

1 What do we do in Economic?

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

C. Hurtado (UIUC - Economics) Game Theory 1 / 31 What do we do in Economic? What do we do in Economic?

What makes a theoretical model ”economics” is that the concepts we are analyzing are taken from real life. Through the investigation of these concepts, we indeed try to understand reality better, and the models provide a language that enables us to think about economic interactions in a systematic way. We do not view economic models as an attempt to describe exactly the world, or to provide tools for predicting the future. Although we will be studying formal concepts and models, they will always be given an interpretation. An economic model diﬀers substantially from a purely mathematical model in that it is a combination of a mathematical model and its interpretation.

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The word ”model” sounds more scientific than ”fable” or ”fairy tale”, but there is not much difference between them. The author of a fable draws a parallel to a situation in real life and has some moral he wishes to impart to the reader. 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 the fable’s advantage. Being something between fantasy and reality, a fable is free of extraneous details and annoying diversions. In this unencumbered state, we can clearly discern what cannot always be seen from the real world. On our return to reality, we are in possession of some sound advice or a relevant argument that can be used in the real world. We do exactly the same thing in economic theory. Thus, a good model in economic theory, like a good fable, identifies a number of themes and elucidates them. We perform thought exercises that are only loosely connected to reality and have been stripped of most of their real-life characteristics. However, in a good model, as in a good fable, something significant remains.

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Rationality forms the basis of decision-making in the neoclassical school. Decision-makers are optimizers, given the constraints they ﬁnd themselves in. Rationality assumes that decision-makers maximize things that give them happiness and minimize things that give them pain. 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 situation where they do not want deviate, the economic system reaches a stable outcome: ’equilibrium’.

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Assumes the existence of many buyers and many sellers. 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 neither inﬂuenced by another agent, nor does it inﬂuence that of another agent. Independent and individual decision-making under perfect competition implies each decision-maker tries to do the best they can irrespective of what other decision-makers are doing. (really?) Perfect competition is a theoretical extreme. Like the ideal human body temperature of 98.4 degrees Fahrenheit it almost never exists. It is used as a benchmark to explain deviations from this ’perfect’ world.

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1 What do we do in Economic?

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

C. Hurtado (UIUC - Economics) Game Theory 6 / 31 What is Game Theory? What is Game Theory?

Branch of applied mathematics and economics that studies strategic situations where there are several players, with different goals, whose actions can affect one another. A game is any situation where multiple players can affect the outcome, a player is a stakeholder, a move or option is an action a player can take and, at the end of the game, the payoff for each player is the outcome. The value of game theory lies in understanding the interactions and likely outcomes when the end result is dependent on the actions of others who have potentially conflicting motives. Game theory is mainly used in economics, political science, and psychology, as well as logic, computer science, and biology. The subject first addressed zero-sum games, such that one person’s gains exactly equal net losses of the other participant or participants. Today, game theory applies to a wide range of behavioral relations: the study of decision science, including both humans and non-humans.

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1 What do we do in Economic?

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

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1713 In a letter dated 13 November 1713 Francis Waldegrave provided the first known, minimax mixed strategy solution to a two-person game. Waldegrave wrote the letter, about a two-person version of a card game, to Pierre-Remond de Montmort who in turn wrote to Nicolas Bernoulli, including in his letter a discussion of the Waldegrave solution. 1838 The French economist Antoine Augustine Cournot discussed a duopoly where the two duopolists set their output based on residual demand. 1871 In the first edition of his book The Descent of Man, and Selection in Relation to Sex Charles Darwin gives the first (implicitly) game theoretic argument in evolutionary biology: If births of females are less common than males, females can expect to have more offspring. Thus parents genetically disposed to produce females tend to have more than the average numbers of grandchildren, hence, female births become more common. As the 1:1 sex ratio is approached, the advantage associated with producing females dies away.

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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? 1921 Emile Borel published four notes on strategic games and an erratum to one of them. Borel gave the first modern formulation of a mixed strategy along with finding the minimax solution for two-person games with three or five possible strategies. Initially he maintained that games with more possible strategies would not have minimax solutions, but by 1927, he considered this an open question as he had been unable to find a counterexample. 1928 John von Neumann proved the minimax theorem. It states that every two-person zero-sum game with finitely many pure strategies for each player is determined, ie: when mixed strategies are admitted, this variety of game has precisely one individually rational payoff vector.

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1934 R.A. Fisher independently discovers Waldegrave’s solution to the card game. Fisher reported 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? 1944 Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern is published. As well as expounding two-person zero sum theory this book is the seminal work in areas of game theory such as the notion of a cooperative game, with transferable utility (TU), its coalitional form and its von Neumann-Morgenstern stable sets. It was also the account of axiomatic utility theory given here that led to its wide spread adoption within economics.

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Von Neumann and Morgenstern formally laid the foundations of Game Theory as a branch of applied mathematics. However, their eﬀort failed to generate stir for two reasons. 1. The Theory of Games and Economic Behavior was based on the notion of zero-sum games. These are known as ’Games of Conﬂict’ or ’Non-Cooperative Games’. Most games in social sciences are non-zero-sum games. 2. It did not establish how equilibrium in games of interdependent decision-making would arise. The world did not have to wait too long for this solution.

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1950 Contributions to the Theory of Games I, H. W. Kuhn and A. W. Tucker eds., published. 1950 In January 1950 Melvin Dresher and Merrill Flood identiﬁed a game where it is in the best interest for players to cooperate, but individual self-interest invokes them to not cooperate. The game reaches a bad equilibrium that is inferior to a superior outcome that could have been reached and was available. This phenomenon was canonized by Albert Tucker. In the summer of 1950 Tucker was at Stanford University. He was working on a problem in his room when a graduate student of psychology knocked and asked what he was doing. The answer was short: game theory. ”Why don’t you explain to us in a seminar”? Tucker used his now famous example of two thieves who were put into separate cells and asked the same question by the judge. Tucker christened the phenomenon as The Prisoners’ Dilemma.

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1953 Extensive form games allow the modeler to specify the exact order in which players have to make their decisions and to formulate the assumptions about the information 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 is currently used, and also some basic theorems pertaining to this class of games.

1953 In four papers between 1950 and 1953 John Nash (1928 - 2015) made seminal contributions 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 to Princeton 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. The number 28 went on to become a superstitious number at Princeton. Von Neumann solved the mini-max theorem in 1928. Nash was born in 1928. Nash’s doctoral dissertation was 28 pages.

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1953 In two papers, Equilibrium Points in N-Person Games (1950) and Non-cooperative Games (1951), Nash proved the existence of a strategic equilibrium for non-cooperative games -the Nash equilibrium- and proposed the ”Nash program”, in which he suggested approaching the study of cooperative games via their reduction to non-cooperative form. This was the missing link to Von Neumann and Morgenstern’s magnum opus!! 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. 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. The intuition behind Nash’s equilibrium is very simple: If all rational economic agents in a system are trying to do the best they can, assuming the others are doing the same, the economic system must be in equilibrium such that no single agent will want to unilaterally deviate from their position. A Nash equilibrium does not necessarily imply that a game will reach a solution that is the best possible solution for all players in the game.

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1951 George W. Brown described and discussed a simple iterative method for approximating solutions of discrete zero-sum games in his paper Iterative Solutions of Games by Fictitious Play. 1953 The notion of the Core as a general solution concept was developed by L. S. Shapley (Rand Corporation research memorandum, Notes on the N-Person Game III: 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 of allocations that cannot be improved upon by any coalition. 50’s Near the end of this decade came the first studies of repeated games. The main result to appear at this time was the Folk Theorem. This states that the equilibrium outcomes in an infinitely repeated game coincide with the feasible and strongly individually rational outcomes of the one-shot game on which it is based. Authorship of the theorem is obscure.

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Identification and formalization of the Prisoners’ Dilemma by Flood, Dresher and Tucker had important implications in economic theory and the social sciences. It provided a rigorous explanation to why intervention into markets can be justified in the presence of public goods that are jointly consumed once provided, but almost impossible to finance through voluntary contribution. This phenomenon formed the basis to why players in a game find a mutual interest to cooperate if a Prisoners’ Dilemma game is played more than once. This lead to what is known as Trigger Strategies. Anatol Rapoport (1911-2007) proposed a simple strategy called Tit For Tat in a repeated Prisoners’ Dilemma tournament. The Tournament was held by the political scientist from Michigan University, Robert Axelrod. Rapoport proposed, each player cooperates with the other player in a repeated Prisoners’ Dilemma as long as the other player does the same. If one player defects in one round, the game ends with the other player applying the Trigger of Tit For Tat by no cooperation and ending the game.

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Variants of Tit for Tat are found in the Grim Trigger introduced by James Friedman (1971); the Trembling Hand Grim Trigger introduced by Michael Taylor in 1987. The Prisoners’ Dilemma in n-person games provided a basis to why individual optimization with a ﬁxed common resource may lead to depletion of the resource over time. This is now known as the Tragedy of the Commons after Garrett Hardin in 1968. The Tragedy of the Commons formed the basis of analyzing global phenomena like Global Warming. Elinor Ostrom (1933-2012) and Oliver Williamson extended the Tragedy of the Commons to discuss the commons and economic governance.

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Reﬁnement of the Nash Equilibrium: The development of Game Theory from the 1950s was based on the limitations of the Nash Equilibrium. The Nash Equilibrium argued that a game can have multiple equilibria (solutions). It did not provide how to identify the set of multiple equilibria and which one to choose as the more probable equilibrium. Thomas Schelling introduced two terms Focal Point (the logical outcome from information from outside a game) and Credible Commitment (sending a signal that a player will commit to a certain actions). - The Focal Point provided a basis for behavioural sciences why players prefer one set of equilibrium to another. It also helped narrow down probable solutions from a larger set. - The Credible Commitment (Threat) gave an added edge to explain previous models like the Stackelberg model in a sequential game setting.

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The Nash Equilibrium analyzed non-cooperative static games. Problems arose with ﬁnding solutions to dynamic games that were sequential or multi stage in nature. The German Game Theorist Reinhard Selten introduced sub-game perfect equilibrium. This equilibrium concept identiﬁes the Nash equilibrium that is consistent with each sub-game in the whole game. Another solution with games that have many decision nodes was introduced by the Israeli-American Game Theorist Robert Aumann. The intuition is simple. It looks at where the game ended. Then it works backward to the beginning of the game. This solution method was termed Backward Induction.

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A further development in the Nash Equilibrium came through the works of the Hungarian-American Game Theorist John Harsanyi. The Nash equilibrium was based on games of complete information. In economic theory, this is known as games under symmetric information when all players in a game make decisions based on having the same set of information. By the 1970s, economists like George Akerlof, Michael Spence and Joseph Stiglitz started analyzing decision-making under asymmetric information or games under incomplete information. Harsanyi had by that time refined the Nash Equilibrium under Bayesian Probability. This enabled the analysis of games where different players have different sets of information about themselves. Harsanyi’s refinement was a blessing for economists who were battling with asymmetric information.

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Goodbye to Rationality: By the 1960s and the 1970s, psychologists studying decision-making, started to challenge the rationality assumption upon which the neoclassical school of economic theory and Game Theory was based. One of the pioneers was Herbert Simon (1916- 2001). He introduced the notion of Bounded Rationality. When individuals make decisions they posses limited information, limited cognitive ability, and limited time within which to make decisions. With the development of bounded rationality, economic theory slowly started to become a behavioural science.

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Behavioral game theory has three ingredients: First: mathematical theories of how ethics and morality aﬀect decision-making. This then aﬀects how individuals or groups bargain and trust each other. Second: how cognitive limitations in the brain restrict the number of steps needed to calculate an optimum decision. Third: how individuals learn from own experience and evolutionary experience and adjust to their decision-making.

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Robert Rosenthal (b 1933) first introduced the Centipede Game in 1981. Two players take turns choosing either to take a slightly larger share of a slowly increasing pot, or to pass the pot to the other player. Based on sub-game perfect equilibrium and backward induction, the rational outcome of the game would be for the player making the first decision to take the entire pot in the first round. However, empirical evidence suggested otherwise. Players tend to partially cooperate so the pot becomes larger as the game proceeds. This empirical outcome challenged the rationality assumption of Game Theory.

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The Ultimatum Game was introduced in 1982 by Werner Guth, Rolf Schmittberger, and Bernd Schwarze. It was later developed by Martin Nowak, Karen Page, Karl Sigmund in 2000 to explain fairness and reasoning in decision-making. In the Ultimatum Game, two players decide how to divide a sum of money. The first player proposes how to divide the sum between the two players. The second player can either accept or reject this proposal. If the second player rejects, neither player receives anything. If the second player accepts, the money is split according to the proposal. The game is played only once. This game has also been termed as the Dictator Game since the first player acts as a dictator declaring their terms under which the sum of money shall be split between the players. This game challenges the rationality assumption. Rationality would suggest the proposer keeps all the money and does not offer anything to the other player. Empirical evidence suggests ethics and morality and evolutionary behaviour leads to some degree of fairness in the splitting of the money.

C. Hurtado (UIUC - Economics) Game Theory 25 / 31 The Theory of Rational Choice On the Agenda

1 What do we do in Economic?

2 What is Game Theory?

3 A Brief History of Game Theory

4 The Theory of Rational Choice

C. Hurtado (UIUC - Economics) Game Theory 26 / 31 The Theory of Rational Choice The Theory of Rational Choice

This theory is that a decision-maker chooses the best action according to her preferences, among all the actions available to her. Her ”rationality” lies in the consistency of her decisions when faced with different sets 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 subset1 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’s preferences. - The set A could, for example, be the set of bundles of goods that the decision-maker can possibly consume; given her income at any time, she is restricted to choose from the subset of A containing the bundles she can afford.

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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 actions as equally desirable (is ”indifferent between the actions”). - Rational Choice (Transitivity): We assume further that these preferences are consistent in the sense that if the decision-maker prefers the action a to the action b, and the action b to the action c, then she prefers the action a to the action c. - Note that we do not rule out the possibility that a person’s preferences are altruistic in the sense that how much she likes an outcome depends on some other person’s welfare. - We can ”represent” the preferences by a payoff function, which associates a number with each action in such a way that actions with higher numbers are preferred. a ≺ b ⇐⇒ u(a) < u(b)

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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 elements that are too ”close” to be distinguishable. The theory of rational choice is that in any given situation the decision-maker chooses the member of the available subset of A that is best according to her preferences. The action chosen by a decision-maker is at least as good, according to her preferences, as every other available action.

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Exercise 1. (Altruistic preferences) Person 1 cares both about her income and about person 2’s income. Precisely, the value she attaches to each unit of her own income is the same as the value she attaches to any two units of person 2’s income. 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 second component is person 2’s income? Give a payoff function consistent with these preferences. Is that payoff unique? Exercise 2. (Alternative representations of preferences) Adecision-maker’s preferences over the set A = a, b, c are represented by the payoff function u forwhich u(a) = 0, u(b) = 1, and u(c) = 4. Are they also represented by the function v for which v(a) = -1, v(b) = 0, and v(c) = 2? How about the function w for which w(a) = w(b) = 0 and w(c) = 8?

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Exercise 3. Deﬃne x ∼ y ⇐⇒ x y and y x. Denote the set of acctions by X. Deﬁne I(x) to be the set of all y in X for which y ∼ x. Show that the set (of sets!) I(x)|x ∈ X is a partition of X. That is: For all x and y, either I(x) = I(y) or I(x) ∩ I(y) = ∅. For every x ∈ X, there is y ∈ X such that x ∈ I(y).

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