# Game Theory Through Examples

Erich Prisner
Edition: 1
Pages: 308
https://www.jstor.org/stable/10.4169/j.ctt6wpwgj

1. Front Matter
(pp. i-vi)
(pp. vii-xv)
3. Preface
(pp. xvi-xx)
4. CHAPTER 1 Theory 1: Introduction
(pp. 1-3)

Every child understands what games are. When someone overreacts, we sometimes say “it’s just a game.” Games are often not serious.Mathematicalgames, 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...

5. CHAPTER 2 Theory 2: Simultaneous Games
(pp. 4-18)

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...

6. CHAPTER 3 Example: Selecting a Class
(pp. 19-24)

All students in Soap College have to enroll either in FRE100 or ITA100 in their second semester. We assume that every student prefers one of them. There is a group of students, however, the “drama queens and kings”, who have strong likes and dislikes between their members, and for whom it is most important to be in a class with as many as possible of drama queens and kings they like and with as few as possible they dislike. A measure of this is the difference between the number of liked and the number of disliked drama queens or kings...

7. CHAPTER 4 Example: Doctor Location Games
(pp. 25-33)

In this chapter and its sibling on Restaurant Location Games we discuss the simultaneous versions of different location games played on undirected graphs.

Like a digraph, an undirected graph consists of vertices. But instead of being connected by arcs with directions, vertices are connected by undirected edge that can be used both ways. Usually they are drawn with curves connecting the vertices. Vertices connected by an edge are called adjacent. Edges may cross, as in the graph in Figure 4.1, where there is an edge connecting vertices 4 and 5, and one connecting vertices 3 and 6, but none connecting...

8. CHAPTER 5 Example: Restaurant Location Games
(pp. 34-41)

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.

Student Activity Play the...

9. CHAPTER 6 Using Excel
(pp. 42-46)

During my senior year in high school I became interested in game theory. At that time I also learned my first programming language, FORTRAN, in a summer course at the University of Stuttgart. The first game I analyzed with computer help was simultaneous QUATRO-UNO described in Project 8. Since then I have believed that analyzing games formally may give new insights only if the game is sufficiently complex, and that for analyzing complex games technology is required. As evidence, this book is exhibit A.

Let’s move from my private belief to the history of science and technology. Von Neumann and...

10. CHAPTER 7 Example: Election I
(pp. 47-52)

The president of the USA is elected by electors from all 50 states. All the electoral votes from a state go to the most popular candidate in that state. If one week before the election a candidate knows that she is behind in a few states, and leading in others, what would be a good strategy for the remaining time? Concentrating on those states where she is behind, or accepting that they are lost and concentrating on others? The decision will depend on several factors, including whether the state can still be turned, and on the size of the state....

11. CHAPTER 8 Theory 3: Sequential Games I: Perfect Information and no Randomness
(pp. 53-69)

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...

12. CHAPTER 9 Example: Dividing A Few Items I
(pp. 70-76)

We assume that the items may have different values to Ann and Beth, that the players know how they both value each item, and that the game’s total value to a player is the sum of the values of the items she got. Both players want to get as much value as possible.

Let us label the items as itemC, D,. . . and leta(C)andb(C)denote the values of itemCfor Ann and Beth, and so on.

Isn’t the way how to play these games obvious? Wouldn’t each player choose the most valuable remaining...

13. CHAPTER 10 Example: Shubik Auction I
(pp. 77-79)

You are probably familiar with English auctions, where players bid for an item. The one with the highest bid gets the item, and pays his or her bid for it. There are many versions dealing with the details, for example whether bidding increments are required, or whether the players must bid in a special order (usually they do not have to).

An English auction is easy to analyze. A player bids as long as the bid is below the worth of the item to him or her, but does not go above that.

In an attempt to make more money...

14. CHAPTER 11 Example: Sequential Doctor and Restaurant Location
(pp. 80-85)

In real life, where rules are not always carved in stone, a player in a simultaneous 2-player game may be able to move a little earlier than the other, or delay slightly until he or she sees what the other player played. If moving earlier is not possible, sometimes you can announce the move you will play, to make a commitment. If the other player believes you, it is as if you have moved first. The simultaneous game is transformed into a sequential game.

There are two roles in the sequentialization of a two-player game, moving first or moving second,...

15. CHAPTER 12 Theory 4: Probability
(pp. 86-98)

Often games contain random features. In poker and other card games, the first move is made not by the players but by the dealer who deals the cards randomly. Outcomes depend on the players’ moves and on random elements. Even some completely deterministic games, like ROCK-SCISSORS-PAPER, are best played using some random device. Accordingly, we need to discuss the theory of probability.

Much mathematical reasoning is concerned with trying to predict outcomes. If I create a sphere of radius 20cm, how large will its surface area be? If I combine 20ml of a 20% acid solution and 30ml of a...

16. CHAPTER 13 France 1654
(pp. 99-101)

1654, Blaise Pascal and Pierre de Fermat exchanged a series of letters. At that time Fermat was 53 years old and a judge in Toulouse. He did mathematics in his spare time, with extraordinary success. A former child prodigy, Pascal was 31 years old and living in Paris. In the letters they discuss a problem posed to Pascal by the Chevalier de Méré, a frequent gambler. Pascal made an attempt at a solution and wrote it to Fermat, probably for confirmation but maybe also for help. We can only guess, since the letter is lost.

Developing scientific ideas in letters...

17. CHAPTER 14 Example: DMA Soccer I
(pp. 102-109)

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...

18. CHAPTER 15 Example: Dividing A Few Items II
(pp. 110-120)

We don’t have a game yet, only the description of a problem.

The most important assumption, used in Section 15.4, is that the daughters are so close to each other that they can estimate the value of each item to their sister. The more complicated incomplete information situation where this is not the case is briefly discussed in Section 15.5.

You could distribute the items yourself. Since you don’t know about the values of the items to your daughters, it may seem fairest to give about the same number of items to each. However, the outcome is not fair if...

19. CHAPTER 16 Theory 5: Sequential Games with Randomness
(pp. 121-128)

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.

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...

20. CHAPTER 17 Example: Sequential Quiz Show I
(pp. 129-134)

Pre-Class Activity: Every student should bring a hard multiple-choice question with five choices for answers from another class.

Student Activity Play the game several times, using the questions brought to class.

It is easy to play the game well if you know the right answer: give it when it is your turn. In the remainder of this chapter we will show how to play the game if the players don’t know the right answer. We assume that the question is so difficult that the players don’t know which choice is right, nor can they rule out any of them of...

21. CHAPTER 18 Las Vegas 1962
(pp. 135-138)

In the 50s and 60s, Las Vegas was different from what it is now. In the 50s it still had a Wild West flavor: No carpets, no dress code, cowboy boots and hats worn in the casinos. In the 60s, when the Mafia took over many of the casinos, those on the strip became more elegant. Alcohol, illegal drugs, and prostitution were never far away, nor were cheating and violence.

Las Vegas then was a world far away from the academic world except for some mathematicians’ interest in gambling. As discussed in Chapter 13, Probability theory started from questions about...

22. CHAPTER 19 Example: Mini Blackjack and Card Counting
(pp. 139-148)

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...

23. CHAPTER 20 Example: Duel
(pp. 149-155)

In this chapter we will discuss some cruel-looking games that model a 19th Century duel. Since dueling is a male game, Ann’s and Beth’s brothers Adam and Bob play it.

The sum of the payoffs is zero if nobody get hurt, but negative otherwise. Duel is not a game you want to play! By changing the payoffs we would be able to create other variants.

Student Activity Play ten rounds of the game in the applet Duel111 to learn how to play.

We have a sequential game with randomness. The extensive form, together with its backward induction analysis, is in...

24. CHAPTER 21 Santa Monica in the 50s
(pp. 156-158)

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...

25. CHAPTER 22 Theory 6: Extensive Form of General Games
(pp. 159-168)

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...

26. CHAPTER 23 Example: Shubik Auction II
(pp. 169-175)

In this chapter we look at this simultaneous game with randomness, and we discuss connections to games with nonperfect and incomplete information. This is a continuation of Chapter 10, where we saw that knowing in advance the maximum number of moves results in a disappointing optimal solution, where the player who will not have the last move will not even start bidding. What happens if the number of bidding rounds is finite but unknown? Or if the number of rounds is finite, but after every move the game could randomly end?

In SHUBIK AUCTION, the player with the last move...

27. CHAPTER 24 Theory 7: Normal Form and Strategies
(pp. 176-184)

In everyday language we distinguish between strategic and tactical behaviors. While tactical reasoning concerns what to do in a particular situation, the strategic point of view considers the whole picture and provides a long-term plan. In game theory, a strategy spans the longest possible time horizon—it is a recipe telling the player what to do in any possible situation. Since a situation translates into an information set in games with imperfect information, a pure strategy for a player lists all information sets in which the corresponding player has to move, together with rules on how to move in each...

28. CHAPTER 25 Example: VNM POKER and KUHN POKER
(pp. 185-192)

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.

The two families of games we describe and partially analyze in this chapter are classics. What I call...

29. CHAPTER 26 Example: Waiting for Mr. Perfect
(pp. 193-198)

WAITING FOR MR. PERFECT is played by one, two, or more players. It has awards, and every player knows what types there are, and how many of each type are available. There is a fixed number of rounds, and there are more awards than rounds. At the start of a round, one of the awards is selected at random. Players simultaneously indicate if they are interested in the award. The award is then given randomly, with equal probability, to one of those who expressed interest. Players who have won an award are out of the game in future rounds.

There...

30. CHAPTER 27 Theory 8: Mixed Strategies
(pp. 199-211)

Student Activity Play ROCK-SCISSORS-PAPER against copro-robot Tarzan in the CoproRobotla applet until you have won three times more than Tarzan.

We have seen that Nash equilibria are likely outcomes of games. What happens in games without a Nash equilibriumin pure strategies? Even simple games like ROCK-SCISSORS-PAPER do not have a Nash equilibrium. In games with pure Nash equilibria players want to communicate their strategies to the other player(s) before the game, but in ROCK-SCISSORS-PAPER it is crucial to leave your opponent in the dark. You want to surprise your opponent, and that may be best achieved by surprising yourself. This...

31. CHAPTER 28 Princeton in 1950
(pp. 212-214)

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...

32. CHAPTER 29 Example: Airport Shuttle
(pp. 215-219)

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...

33. CHAPTER 30 Example: Election II
(pp. 220-224)

This chapter is a continuation of Chapter 7, where we described versions of the games ELECTION(c, d, e|c₁, d₁, e₁| a, b). We did not discuss the many versions that don’t have pure Nash equilibria, but now mixed strategies allow us to take a different tack. Since we can model these games as two-person zero-sum games, von Neumann’s and Robinson’s theorems apply, and the mixed Nash equilibria are meaningful. We will use the Excel sheet Election2.xlsm, where the payoff matrix is calculated automatically, and where we implement Brown’s fictitious play method to approximate mixed Nash equilibria.

We lay a foundation...

34. CHAPTER 31 Example: VNM POKER(2, r, m, n)
(pp. 225-230)

In this chapter we use mixed strategies to look at versions of VNM POKER(2,r, m, n). It gives exercise in working with parameters, manipulating algebraic expressions, and graphing equations.

We play VNM POKER(2, r, m, n) withrcards of value 1 andrcards of value 2, with initial bets ofmand raised bets ofn. We assumer≥ 2. The extensive form of VNM POKER(2, 4, 2, 3) is shown in Figure 31.1.

Class Activity In the applet VNMPoker2, select values ofmandnand a computer opponent (Chingis, Grandpa, or Lucky) and...

35. CHAPTER 32 Theory 9: Behavioral Strategies
(pp. 231-236)

A pure strategy is like instructions written in a notebook that an overprotective father gives his child before the child goes to middle school in a big city. They should describe clearly and precisely what to do in each every situation the child might encounter. Many will not occur, because of chance or because the other persons don’t make the decisions that lead to them, or because the child makes decisions that prevent them from occurring. The child can avoid some situations but not others. The strategy prepares the child for every possible situation.

Continuing the metaphor, we can think...

36. CHAPTER 33 Example: Multiple-Round Chicken
(pp. 237-243)

In October 1962, US surveillance discovered that the USSR was about to install offensive atomic missiles in Cuba. For one week President John F. Kennedy and his advisors kept the knowledge secret and discussed what to do. It has been reported that in these meetings the Secretary of Defense, Robert McNamara, outlined three options: trying to solve the problem politically, continuing the surveillance and starting a blockade, or undertaking military action against Cuba. A week after the discovery, when it released information about the crisis, the US announced it would undertake a blockade, which it called a “quarantine”. As Russian...

37. CHAPTER 34 Example: DMA Soccer II
(pp. 244-251)

Coaches of team sports like soccer supervise the training of their teams, and they play in the sense that they select the players and the system their team will use. Besides trying to motivate their teams, coaches also have the important task of reacting to what they see by substituting players or changing the system.

In a Champion’s League soccer game between Bolton Wanderers and Bayern Munich in November 2007, Munich was leading 2-1 when their coach Ottmar Hitzfeld substituted for two key players on his team, Frank Ribery and Bastian Schweinsteiger, taking them out. After that, the Bolton Wanderers...

38. CHAPTER 35 Example: Sequential Quiz Show II
(pp. 252-258)

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...

39. CHAPTER 36 Example: VNM POKER(4, 4, 3, 5)
(pp. 259-263)

This chapter demonstrates the difficulties one may face when looking for mixed Nash equilibria in larger examples. Moreover we see that though one player’s Nash equilibrium mix may draw out many more strategies than the other player’s corresponding Nash equilibrium mix, playing the mix may still be worthwhile in two-player zero-sum games.

Class Activity In the applet VNMPoker4, select valuesmandn, one of the computer opponents, and play 30 rounds.

This example is more complex than the caseS= 2 discussed in Chapter 31. An analysis of the family of games gets too complex for general parameters...

40. CHAPTER 37 Example: KUHN POKER(3, 4, 2, 3)
(pp. 264-267)

Student Activity Play ten rounds of KUHNPOKER(3; 4; 2; 3) in the applet KuhnPoker3 to refresh your familiarity with the game.

In Chapter 25 we gave a description of the games and provided extensive forms. We found that some pure and reduced pure strategies are weakly dominated and described the normal form for a few cases. Although some small cases of the games had pure Nash equilibria, we saw this is not true for KUHN POKER with S = 3. In this chapter we find a Nash equilibrium in mixed strategies for KUHNPOKER(3; 4; 2; 3). In our discussion we...

41. CHAPTER 38 Example: End-of-Semester Poker Tournament
(pp. 268-276)

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.

In Spring 2011, the students and teacher submitted the robots in Table 38.1.

Since the string of twelve numbers uniquely determines the robot’s behavior, we call it the DNA of the robot. To get sixteen players for a knockout tournament, three more robotswere added: two identical Random robots, which always decide randomly with...

42. CHAPTER 39 Stockholm 1994
(pp. 277-279)

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. TheNew York Timeshad 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...

43. Bibliography
(pp. 280-283)
44. Index
(pp. 284-287)