Introduction to 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. C. Hurtado (UIUC - Economics) Game Theory 2 / 31 What do we do in Economic? What do we do in Economic? 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. C. Hurtado (UIUC - Economics) Game Theory 3 / 31 What do we do in Economic? As if Rationality 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’. C. Hurtado (UIUC - Economics) Game Theory 4 / 31 What do we do in Economic? Perfect Competition is a Benchmark 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. C. Hurtado (UIUC - Economics) Game Theory 5 / 31 What is 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 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. C. Hurtado (UIUC - Economics) Game Theory 7 / 31 A Brief History of 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 8 / 31 A Brief History of Game Theory A Brief History of Game Theory 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. C. Hurtado (UIUC - Economics) Game Theory 9 / 31 A Brief History of Game Theory A Brief History of Game Theory 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. C. Hurtado (UIUC - Economics) Game Theory 10 / 31 A Brief History of Game Theory A Brief History of Game Theory 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. C. Hurtado (UIUC - Economics) Game Theory 11 / 31 A Brief History of Game Theory A Brief History of Game Theory 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. C. Hurtado (UIUC - Economics) Game Theory 12 / 31 A Brief History of Game Theory A Brief History of Game Theory 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.