Moving from the Extensive Form and Strategic Form 39
� SITUATION: THE SCIENCE 84 GAME
The magazine Science 84 came up with the idea of running the following con- test for its readership: Anyone could submit a request for either $20 or $100. If no more than 20% of the submissions requested $100, then everybody would receive the amount he or she requested. If more than 20% of the sub- missions asked for $100, then everybody would get nothing.
The set of players is the set of people who are aware of the contest. The strategy set for a player is made up of three elements: do not send in a request, send in a request for $20, and send in a request for $100. Let us suppose that each player’s payoff is the amount of money received, less the cost of submit- ting a request, which we’ll assume is $1 (due to postage and the time it takes to write and mail a submission).
In writing down player i’s payoff function, let x denote the number of play- ers (excluding player i) who chose the strategy send in a request for $20 and y denote the number of players (excluding player i) who chose send in a request for $100. Then player i’s payoff function is:
0 if i chooses do not send in a request
19 if i chooses send in a request for $20 and � .2 y
x � y � 1
99 if i chooses send in a request for $100 and � .2 y � 1
x � y � 1
�1 if i chooses send in a request for $20 and .2 � y
x � y � 1
�1 if i chooses send in a request for $100 and .2 � y � 1
x � y � 1
For example, if player i requested $20, and no more than 20% of the submis-
sions requested $100 (i.e., � .2), then she receives $20 from Science 84,
from which we need to subtract the $1 cost of the submission. Although it would be great to know what happened, Science 84 never ran
the contest, because Lloyd’s of London, the insurer, was unwilling to provide insurance for the publisher against any losses from the contest.
2.6 Moving from the Extensive Form and Strategic Form FOR EVERY EXTENSIVE FORM GAME, there is a unique strategic form representation of that game. Here, we’ll go through some of the preceding examples and show how you can derive the set of players (that one’s pretty easy), the strategy sets, and the payoff functions in order to get the corresponding strategic form game.
� SITUATION: BASEBALL, II
Consider the Baseball game in Figure 2.2. The strategy set of the Orioles’ man- ager includes two elements: (1) Substitute Gibbons for Lopez and (2) retain Lopez. As written down, there is a single information set for the Yankees’ man- ager, so his strategy is also a single action. His strategy set comprises (1) sub- stitute Johnson for Rivera and (2) retain Rivera. To construct the payoff ma- trix, you just need to consider each of the four possible strategy profiles and determine to which terminal node each of them leads.
y x � y � 1
⎞ ⎪ ⎪ ⎪ ⎬ ⎟ ⎪⎪ ⎠
40 CHAPTER 2: BUILDING A MODEL OF A STRATEGIC SITUATION
If the strategy profile is (retain Lopez, retain Rivera), then the payoff is 2 for the Orioles’ manager and 2 for the Yankees’ manager, since Lopez bats against Rivera. The path of play, and thus the payoffs, are the same if the profile is in- stead (retain Lopez, substitute Johnson), because substitute Johnson means “Put in Johnson if Gibbons substitutes for Lopez”. Since the latter event doesn’t occur when the Orioles’ manager chooses retain Lopez, Johnson is not substi- tuted. When the strategy profile is (substitute Gibbons, retain Rivera), Gibbons bats against Rivera and the payoff pair is (3,1), with the first number being the payoff for the Orioles’ manager. Finally, if the strategy profile is (substitute Gibbons, substitute Johnson), Gibbons bats against Johnson and the payoff pair is (1,3). The payoff matrix is then as depicted in FIGURE 2.13.