Does the game have to be factually accurate?
If our objective in formulating and then solving a game is to understand be- havior, then what matters is not what is factually or objectively true, but rather what is perceived by the players. Their behavior will be driven by their pref- erences and what they believe, whether those beliefs are in contradiction with reality or not. Thus, a game ought to represent players’ environment as it is perceived by them.
48 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
If a player is a member of a Native American tribe in the 19th century which believes that a tribal leader has magical powers, we need to recog- nize that belief—regardless of whether or not it is true—if we are to under- stand their behavior. Or if a player in the 21st century believes that flying a plane into a building can improve his well-being in the afterlife, then we need similarly to recognize that belief, no matter how wrong or misguided it may be.
Summary When the French Impressionist painter Claude Monet viewed a London building, a French cathedral, or a lily pond in his backyard, he painted, not reality, but his impression of it. Modeling real-life encounters between peo- ple is similarly an art form, though admittedly not one worth framing and hanging on your wall. Real life is complicated, nuanced, and messy, and a social scientist who wants to understand it must distill its essential features if he is to construct a simple and parsimonious model. Doing so requires cre- ativity, insight, and judgment. While game theory cannot bring those attrib- utes to the table, it can provide the tools for the intelligent observer who has such traits to build a model that will shed light on why people do the things they do.
In this chapter, we have reviewed the two frameworks for constructing a game-theoretic model of a strategic situation. An extensive form game uses a tree structure to depict the sequence in which players make decisions and describes the circumstances surrounding those decisions, including the actions available to a player and what the player knows regarding what has happened in the game. That knowledge is represented by an information set which encompasses all those paths in the game that a player is in- capable of distinguishing among. The concept of an information set allows us to model the many different contexts in which decisions are made while we lack relevant facts. An information set can embody a situation of per- fect information, in which a player knows all that has transpired thus far in the game, or one of imperfect information, in which a player has some uncertainty in regard to what other players have done. Key to describing behavior is knowing what players care about, so an extensive form game also describes the well-being, or payoff, that a player assigns to an outcome of the game.
A strategic form game has a more concise format than the extensive form game has. A strategic form game is defined by the set of players, the strategy set of each player, and the payoff function of each player. A player’s decision making involves the selection of a strategy from his or her strategy set, where a strategy is a fully specified decision rule for how to play a game. A payoff function tells us how a player evaluates any collection of strategies (one for each of the players in the game).
As will be revealed in the ensuing chapters, crucial to both predicting be- havior and prescribing behavior is knowing what each player knows about the other players, and this knowledge includes what each player believes about the other players. A central underlying assumption is that the game is common knowledge to the players. This means not only that players agree
1. The countries of Oceania and Eurasia are at war.5 As depicted in FIGURE PR2.1, Oceania has four cities—Argula, Betra, Carnat, and Dussel—and it is concerned that one of them is to be bombed by Eurasia. The bombers could come from either base Alpha, which can reach the cities of Argula and Betra, or base Beta, which can reach either Carnat or Dussel. Eurasia decides which one of these four cities to attack. Oceania doesn’t know which one has been selected, but does observe the base from which the bombers are flying. After making that observation, Oceania decides which one (and only one) of its four cities to evacuate. Assign a payoff of 2 to Oceania if it succeeds in evacuating the city that is to be bombed and a payoff of 1 otherwise. Assign Eurasia a payoff of 1 if the city it bombs was not evacuated and a zero payoff otherwise. Write down the extensive form game.
EXERCISES
Exercises 49
Oceania
Argula
Betra
Beta Alpha
Dussel
Carnat
FIGURE PR2.1
2. Player 1 moves initially by choosing among four actions: a, b, c, and d. If player 1 chose anything but d, then player 2 chooses between x and y. Player 2 gets to observe the choice of player 1. If player 1 chose d, then player 3 moves by choosing between left and right. Write down the ex- tensive form of this setting. (You can ignore payoffs.)
on the game that is being played, but also that each player knows what the other players believe about the game, and so forth. Common knowledge is like the perfect circle; it is a concept that does not exist in reality, but nev- ertheless is a useful abstraction for understanding the world within which we live.
50 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
3. Consider a setting in which player 1 moves first by choosing among three actions: a, b, and c. After observing the choice of player 1, player 2 chooses among two actions: x and y. Consider the following three variants as to what player 3 can do and what she knows when she moves: a. If player 1 chose a, then player 3 selects among two actions: high and
low. Player 3 knows player 2’s choice when she moves. Write down the extensive form of this setting. (You can ignore payoffs.)
b. If player 1 chose a, then player 3 selects among two actions: high and low. Player 3 does not know player 2’s choice when she moves. Write down the extensive form of this setting. (You can ignore pay- offs.)
c. If player 1 chose either a or b, then player 3 selects among two ac- tions: high and low. Player 3 observes the choice of player 2, but not that of player 1. Write down the extensive form of this setting. (You can ignore payoffs.)
4. Return to the game involving the U.S. Court of Appeals in Section 2.2. a. Suppose, at the start of the game, it is known by all that Judge Z will
read only the brief of Ms. Hasenpfeffer. Write down the correspon- ding extensive form game. You may exclude payoffs.
b. Suppose, at the start of the game, it was known by all that Judge X would vote first and reveal his vote to Judges Y and Z before they vote simultaneously. Write down the corresponding extensive form game. You may exclude payoffs.
5. The city council is to decide on a proposal to raise property taxes. Suppose Ms. Tuttle is the chair and the Council’s other two members are Mr. Jones and Mrs. Doubtfire. The voting procedure works as follows: Excluding the chair, Mr. Jones and Mrs. Doubtfire simultaneously write down their votes on slips of paper. Each writes either for or against the tax increase. The secretary of the city council then opens the slips of paper and announces the vote tally. If the secretary reports that both slips say for, then the tax increase is implemented and the game is over. If both vote against, then the tax increase is not implemented and, again, the game is over. However, if it is reported that the vote is one for and one against, then Ms. Tuttle has to vote. If she votes for, then the tax increase is implemented, and if she votes against, then it is not. In both cases, the game is then over. As to payoffs, if the tax increase is implemented, then Mrs. Doubtfire and Mr. Jones each receive a payoff of 3. If the tax in- crease proposal fails, then Mrs. Doubtfire has a payoff of 4 and Mr. Jones’s payoff is 1. As for Ms. Tuttle, she prefers to have a tax increase— believing that it will provide the funds to improve the city’s schools—but would prefer not to be on record as voting for higher taxes. Her payoff from a tax increase when her vote is not required is 5, her payoff from a tax increase when her for vote is required is 2, and her payoff from taxes not being increased is zero (regardless of whether or not she voted). Write down the extensive form of the game composed of Ms. Tuttle, Mr. Jones, and Mrs. Doubtfire.
6. Consider a contestant on the legendary game show Let’s Make a Deal. There are three doors, and behind two doors is a booby prize (i.e., a prize of little value), while behind one door is a prize of considerable value, such as an automobile. The doors are labeled 1, 2, and 3. The strategic situation starts when, prior to the show, host Monty Hall selects one of the three doors behind which to place the good prize. Then, during the show, a
Exercises 51
contestant selects one of the three doors. After its selection, Monty opens up one of the two doors not selected by the contestant. In opening up a door, a rule of the show is that Monty is prohibited from opening the door with the good prize. After Monty Hall opens a door, the contestant is then given the opportunity to continue with the door originally selected or switch to the other unopened door. After the contestant’s decision, the re- maining two doors are opened. a. Write down an extensive form game of Let’s Make a Deal up to (but
not including) the stage at which the contestant decides whether to maintain his original choice or switch to the other unopened door. Thus, you are to write down the extensive form for when (1) Monty Hall chooses the door behind which the good prize is placed, (2) the contestant chooses a door, and (3) Monty Hall chooses a door to open. You may exclude payoffs.
b. For the stage at which the contestant decides whether or not to switch, write down the contestant’s collection of information sets. In doing so, denote a node by a triple, such as 3/2/1, which describes the sequence of play leading up to that node. 3/2/1 would mean that Monty Hall put the good prize behind door 3, the contestant initially selected door 2, and Monty Hall opened door 1.
7. For the Iraq War game in Figure 2.10, write down the strategy sets for the three players.
8. For the extensive form game in FIGURE PR2.8, derive its corresponding strategic form.
5
2
2
2
1
1
15
0
0 Player 1
0 Player 2
10
5
c2
a2
a1
b2
b1
d2
c1 d1
20
3
4
1
c2 d2
FIGURE PR2.8
52 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
9. Write down the strategic form game for the extensive form game in FIGURE PR2.9.
Player 1
Player 2
2
1
1
3
4
2
a b c d
x y x y x y x y
2
2
0
6
3
1
4
2
1
5
0
0
FIGURE PR2.9
10. Write down the strategic form game for the extensive form game in FIGURE PR2.10.
11. Three extensive form games are shown in FIGURE PR2.11. State which of them, if any, violate the assumption of perfect recall. Explain your answer.
Player 1
Player 2
Player 3
2
3
4
1
a b c
x y x y r s
1
1
6
0
2
3
1
0
02 1 3 5 6
3
1
2
FIGURE PR2.10
Exercises 53
FIGURE PR2.11
(b)
0
1
1
1
x y x y
c dc d c d c d
0
0
1
0
1
0
1
1
1
0
0
1
2
1
1 1
(c)
0
1
1
1
x y x y
c dc d c d c d
0
0
1
0
1
0
1
1
1
0
0
1
2
1
1
(a)
0
1
1
1
a b
x y x y
c dc d c d c d
0
0
1
0
a b
a b
1
0
1
1
1
0
0
1
2 2
1
1 11
54 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
REFERENCES 1. John Thorn and Pete Palmer, The Hidden Game of Baseball (New York:
Doubleday/Dolphin, 1985). The seasons analyzed were 1974–77.
2. Sun Tzu, The Art of War, translated by Thomas Cleary, Shambhala, 1998. For an examination of Sun Tzu’s writings using the lens of game theory, see Emerson M. S. Niou and Peter C. Ordeshook, “A Game-Theoretic Interpretation of Sun Tzu’s The Art of War,” Journal of Peace Research, 31 (1994), 161–74.
3. J. D. Williams, The Compleat Strategyst (New York: McGraw Hill, 1954), p. 16.
4. This application of game theory was first suggested by Anatol Rapoport, “The Use and Misuse of Game Theory,” Scientific American, 207(6): 108–18.
5. These fictitious countries appear in the George Orwell novel 1984. If dystopia is not your bag, then you can use Freedonia and Sylvania in their place, which are fictional nations from the 1933 Marx Brothers’ movie Duck Soup.