Student Activity Play RANDOM NIM(5,0.5) at least 20 times against the computer in the applet Nim7Rc. Put in the values 5 and 0.5 into the text field before you start. Try to win.

The game is sequential, with two players, but between moves of the players there are the random removals. It is a game that is sequential with randomness, and we discuss them in this chapter.

Extensive Form Extended

To describe and discuss sequential games with some random moves, we will merge the concept of extensive forms of sequential games as described in Chapter 8 with the concept of probability trees discussed in Chapter 12. In addition to the vertices that correspond to positions of the game where a player makes a decision, we need random vertices. There the random moves are performed and there are arcs to other vertices from them. These are the positions that can be reached from these random positions. The arcs are labeled by the appropriate probabilities to reach the other positions from the random position.

Welcome to game theory, the mathematical theory of how to analyze games and how to play them optimally. Although “game” usually implies fun and leisure, game theory is a serious branch of mathematics. Games like blackjack, poker, and chess are obvious examples, but there are many other situations that can be formulated as games. Whenever rational people must make decisions within a framework of strict and known rules, and where each player gets a payoff based on the decisions of all the players, we have a game. Examples include auctions, negotiations between countries, and military tactics. The theory was initiated by mathematicians in the first half of the last century, but since then much research in game theory has been done outside of mathematics.

This book gives an introduction to game theory, on an elementary level. It does not cover all areas of a field that has burgeoned in the last sixty years. It tries to explain the basics thoroughly, so that it is understandable for undergraduates even outside of mathematics, I hope in an entertaining way. The book differs from other texts because it emphasizes examples. The theory is explained in nine theory chapters, and how it is applied is illustrated in twenty-four example chapters, where examples are analyzed in detail. Each example chapter uses tools developed in theory chapters.

Audience

The text can be used for different purposes and audiences. It can be used by students of economics, political science, or computer science. Undergraduate mathematics students should enjoy the book as well and profit from its approach. The book may also be used as a secondary text, or for independent study, as its many concrete examples complement expositions of the theory.

Student Activity Play VNM-Poker in the applet VNMPokerseq13. The description can be found there.

Student Activity Play 2-round WAITING FOR MR. PERFECT in the applet Waiting2b.

Chapter 2 was about simultaneous games. We showed how a matrix representation, the normal form, can be helpful in analyzing a game. In Chapters 8 and 16, which dealt with sequential games with or without randomness, we learned how to describe such games in extensive form, and how these games have a clearly defined solution and (expected) value, which we can compute using backward induction. An important assumption was that we had perfect information about the games.

However, most games are neither simultaneous nor purely perfect-information sequential. They are something in between. Some are essentially sequential with some moves done in parallel, like WAITING FOR MR PERFECT; some are sequential but lack perfect information like VNM-POKER. Perfect information means that every player is aware of all previous moves of all players. Simultaneous moves can be rephrased as sequential moves with non-perfect information–we put them into any order without giving the player who moves second knowledge about the move of the player moving first. In this chapter we will see that the description in extensive form can be applied to any game, even one with imperfect information.

Prerequisites: Chapters 1, 2, and 12.

Imagine you are in a poker tournament and the organizer approaches your chair, taps you on the shoulder, and tells you to pass less often. Would you be amused? In a liberal society, we have become accustomed to the fact that organizations and governments leave us freedom. Of course, there are always rules such as pay your taxes, don't speed, and don't smoke during class, but we want them to be minimal, and we enjoy the freedom of selecting our own moves. In many games, players' behavior can be influenced in subtle ways by those in charge. All they need to do is to change the rules. Increase taxes for flying and the people will more often travel by car or train.

In 1994 FIFA, the World Soccer Association, changed the rules for assigning points in soccer tournaments. Before that, a win counted as 2 points, a draw 1 point, and a loss as 0 points. With the new rule, a win counted 3 points. FIFA's aim was to encourage teams to play more offense, after years of defensive, unattractive play.

Prerequisites: Chapters 1 and 2.

As in in the previous chapter we discuss games on graphs. Their most interesting feature is that they always have pure Nash equilibria.

Let us repeat the definition of an undirected graph, given in Chapter 4. Undirected graphs have vertices, displayed by small circles. Some pairs of vertices are connected by undirected edges. They are drawn as curves connecting the two vertices. Vertices connected by an edge are called adjacent. Adjacent vertices are also called neighbors.

Again we use graphs to model islands. The vertices represent the towns, and the edges the roads between the towns.

Prerequisites: Chapters 8, 12, and 16.

Do you gamble at a casino? Would you? Why not? If your answer is that casino games are designed so that the Casino has better odds, you are only partially right. Blackjack is one of the few casino games where playing optimally may actually win you money in the long run. In the last chapter we mentioned some attempts to analyze the game. For those of you eager to go and bankrupt a casino, you will be disappointed to hear that we will not discuss blackjack in detail. There are two reasons. First, casino blackjack is too complicated to treat in an introductory book. Second, and more important, it has a huge extensive form. A similar game called MINI BLACKJACK has most of the essential ingredients of casino blackjack, and it is simple enough for us to analyze. We will see three aspects of casino blackjack, namely counting cards, playing against the house, and reshuffling, in versions of the game.

The Basic Game

The basic game depends on two parameters a and b:

In his story “Jewish Poker” the writer Ephraim Kishon describes how a man called Ervinke convinces the narrator to play a game called Jewish Poker with him. “You think of a number, I also think of a number”, Ervinke explains. “Whoever thinks of a higher number wins. This sounds easy, but it has a hundred pitfalls.” Then they play. It takes the narrator some time until he realizes that it is better to let Ervinke tell his number first. [K1961] Obviously this is a game that is not fair unless both players play simultaneously.

In this chapter we will start our journey through game theory by considering games where each player moves only once, and moves are made simultaneously. The games can be described in a table (called the game's normal form). Then we discuss approaches that allow the players to decide which move they will choose, culminating with the famous Nash equilibrium.

Normal Form—Bimatrix Description

Imagine you want to describe a simultaneous game. We know that each player has only one move, and that all moves are made simultaneously. What else do we need to say? First, we must stipulate the number of players in the game. Second, we must list for each player all possible moves. Different players may have different roles and may have different options for moves. We assume that each player has only finitely many options. Players simultaneously make their moves, determine the outcome of the game, and receive their payoffs. We need to describe the payoffs for each outcome.

Prerequisites: Chapters 2 and 27.

In Lugano, the hometown of our college, there are two competing shuttle companies that serve the airport in Milano-Malpenso (MXP), which is about 1 hour and 15 minutes away. They have different schedules, but both companies depart from Lugano's railway station. Quite often company A's shuttles are supposed to leave just briefly before company B's. I wondered how many, or actually how few, customers would take the shuttle from company B. Probably only those who arrive at the railway station in the short interval between shuttle A's and shuttle B's departures.

It seems obvious that it matters how the two companies' schedules relate to one another. How should they set up their schedules?

The Simple Model

In 1969, as a small German boy, I watched together with millions of others as Americans set foot on the moon. For me this was an incredible demonstration of the power of science and technology and of US leadership in those fields. It seemed that anything was possible.

I was too young to have heard about the threats of the cold war, about the treatment of the blacks in the US, and the resulting riots, about assassinations of presidents and Martin Luther King, about protests in 1968 in all the western world, and about the Vietnam war, although I must have seen some glimpses of that on my grandma's newly acquired TV. Instead I drew my knowledge about the US from the Big Book on America, which I got from our local library. This book was from the 50s and painted a very optimistic picture of the US, with all people nice, friendly, and rational. Happy (white) families in nice, clean houses in the suburbs, the men driving to offices in huge cars that resembled ships, dressed in white short-sleeved shirts and dark suits, heading to interesting jobs in huge air-conditioned buildings. Nice, functional, modern buildings, like the RAND building in Santa Monica.

In Fall 1994, mathematician John F. Nash received a phone call that he had won a Nobel prize, together with economists John C. Harsanyi and Reinhard Selten. The next day the story filled the newspapers. The New York Times had a headline “Game theory captures a Nobel”. Mathematical journals reported the news weeks or months later.

Hold on a minute! There is no Nobel prize in mathematics! In the early twentieth century Alfred Nobel established prizes only in physics, chemistry, medicine, literature, and peace. Nash worked in none of these disciplines. His award was not one of the original Nobel prizes but was rather the Nobel Memorial Prize in Economic Sciences, which was initiated in 1968 and is sponsored by the Central National Bank of Sweden. Like the original Nobel prizes, it is awarded by the Royal Swedish Academy of Sciences. It is often through this prize that mathematicians hit the front pages of the newspapers and become famous.

Nowadays the academic areas closest to mathematics are probably physics, computer science, and economics. When Greek culture dominated the western world, philosophy was close to mathematics, but the two have separated since then. Astronomy and mathematics were closely related in ancient times, but the close relationship between physics and mathematics was established and strengthened in the 17th century through the development of calculus.

Student Activity Try your luck in applet AppletNim7 against a friend (hit the “Start new with 6” button before you start). Or play the game against the computer in AppletNim7c. Play ten rounds where you start with seven stones. Then play ten rounds where you start with nine stones. Then play ten rounds where you start with eight stones. Discuss your observations.

In this chapter we look at a class of simple games, namely sequential games. They are games where the players move one after another. Among them we concentrate on games of perfect information, where players know all previous decisions when they move. Randomness will be included after the next chapter. We will learn a little terminology, see how to display the games either as a game tree or a game digraph, and how to analyze them using “backward induction” provided the game is finite. We conclude by discussing whether the solution found by backward induction would be what real players would play, by discussing another approach for sequential games, by discussing the special roles two-person zero-sum games play here, and by discussing briefly the well-known sequential games chess, checkers, and tic-tac-toe.

A glimpse into cooperative game theory

Prerequisites: Chapters 16 and 17.

This chapter provides a glimpse into cooperative game theory, which is otherwise not covered. We will investigate what happens in SEQUENTIAL QUIZ SHOW(n, m) if two players cooperate, and share evenly, or according to a formula, a win or loss. In Section 35.1 we look at fixed coalitions. In Section 35.2 we ask which coalitions are most likely to form provided the players in one have to share a win or loss evenly. In Section 35.3 we drop the requirement of having to share evenly. In Section 35.4 we investigate what would be a fair share of the win if all three players work together.

Here is the game description again:

What's a Game?

Every child understands what games are. When someone overreacts, we sometimes say “it's just a game.” Games are often not serious. Mathematical games, which are the subject of this book, are different. It was the purpose of game theory from its beginnings in 1928 to be applied to serious situations in economics, politics, business, and other areas. Even war can be analyzed by mathematical game theory. Let us describe the ingredients of a mathematical game.

Rules Mathematical games have strict rules. They specify what is allowed and what isn't. Though many real-world games allow for discovering new moves or ways to act, games that can be analyzed mathematically have a rigid set of possible moves, usually all known in advance.

Outcomes and payoffs Children (and grown-ups too) play games for hours for fun. Mathematical games may have many possible outcomes, each producing payoffs for the players. The payoffs may be monetary, or they may express satisfaction. You want to win the game.

Uncertainty of the Outcome A mathematical game is “thrilling” in that its outcome cannot be predicted in advance. Since its rules are fixed, this implies that a game must either contain some random elements or have more than one player.

Prerequisites: Chapters 2, 8, 12, 16, 22, and 24.

Finally, a chapter on poker! Now we will learn how to play it well and make a lot of money, right? If this was your thought, I have to disappoint you. Advice on poker is possible, but the game is again too complex for a mathematical analysis. What we will do instead is study several small variants that we can analyze. This chapter will introduce the games, and do some analysis. We will complete the analysis in Chapters 31, 36, and 37.

Description

The two families of games we describe and partially analyze in this chapter are classics. What I call VNM POKER was introduced by von Neumann and Morgenstern in their monograph [VNM1944]. KUHN POKER (in the variant (3, 1, 1, 2)) was introduced by Kuhn in [K1950].

Both games have four parameters, S, r, n, and m, with m < n. There are cards of value from 1 to S in a deck, and each value occurs r times. So there are S · r cards. There are two players, Ann and Beth. Each player randomly gets a card from the deck, looks at it, but doesn't show it to her opponent. The ante is m, meaning that both player put m dollars into the pot at the start of the game. Each player's bet can raise to n if the player puts n − m additional dollars in.

Prerequisites: Chapter 25 and all theory chapters, in particular Chapters 24, 27, and 32.

Each semester my game theory class has a poker tournament, playing either a version of KUHN POKER or a version of VNM POKER. The assignment looks like this.

Class Activity Create your own robot. What twelve numbers would you choose, and why? Discuss your choices with classmates.

The 2001 movie “A Beautiful Mind” focused on game theorist John F. Nash and was a huge success, winning four Academy Awards. It is based on Sylvia Nasar's unauthorized biography with the same title, published in 1998. There is an amazing passage in the book, which unfortunately was left out of the movie. It is where Nash visited John von Neumann in his office to discuss the main idea and result of the Ph.D. thesis he was writing. At the time Nash was a talented and promising mathematics student at Princeton University and von Neumann was a professor at the Institute for Advanced Study in Princeton, a member of a handful of mathematicians of the highest distinction. Albert Einstein was a colleague. According to the book's description of the meeting, based on an interview with Harold Kuhn, the visit was short and ended by von Neumann exclaiming: “That's trivial, you know. That's just a fixed point theorem” [N1998].

Reading the scene, I see the two men, one eager to impress with his theorem, the other seeming to try to not be impressed. Both men were competitive, more than what may be normal for mathematicians. Although few can answer difficult mathematical questions, many can tell right from wrong answers. For instance, solving an equation may be hard, but checking whether a number is a solution is easy. For this reason, contests are much more frequent in mathematics than in other academic areas. In the 16th century, mathematicians challenged each other, posing questions that they hoped would show their superiority. Nowadays, there are many mathematical contests for students, Putnam Exams, national and international Mathematical Olympiads, and others. For adult mathematicians, there is the game of challenging each other with conjectures, and winning by being the first to prove or disprove one.