## Tuesday, September 6, 2011

### More on Glazer and Rubinstein on debates

By the way, I computed the entire set of winning strategies that goes with the decision rule that Glazer and Rubinstein present on p. 255 of Game Theory and Pragmatics.

The rule regards a case with five witnesses numbered 1 through 5. Debater 1 first cites one witness that supports his case, and debater 2 then cites a witness that supports her case. The decision rule then specifies the pairs of arguments after which the listener will guess that debater 2's case reflects the pool of witness better. Using the specific rule on p. 255, the listener does so after these pairs of arguments:
•  <1,2>, <2,3>, <2,5>, <3,4>, <4,2>, <4,5>, <5,1>, <5,4>
A pair such as <5,4> is here read as an indication that the listener will believe debater 2 if debater 1 has cited witness 5, and debater 2 has cited witness 4.

Note that only 8 out of the 5 x 4 = 20 possible histories  favour debater 2. This reflects the fact that the game is sequential, and that debater 2 moves last. She just has an information advantage.

The rule could also be specified by labeling the leaves of a tree. In that case we should read each pair as paths from the root to a leaf, and we should label all these leafs "2" and the rest "1."

Given this decision rule, we can compute the set of arguments E that are winning strategies for debater 1 in a given state s. It turns out that this function E = E(s) has the following values:
• E(<1,1,1,1,1>) = {1, 2, 3, 4, 5}
• E(<1,1,1,1,2>) = {1, 3}
• E(<1,1,1,2,1>) = {1, 2}
• E(<1,1,1,2,2>) = E(<1,1,2,1,2>) = E(<1,1,2,2,1>) = E(<1,1,2,2,2>) = {1}
• E(<1,1,2,1,1>) = {1, 4, 5}
• E(<1,2,1,1,1>) = {3, 5}
• E(<1,2,2,1,1>) = E(<2,1,1,1,2>) = E(<2,2,1,1,1>) = E(<2,2,1,1,2>) = {3}
• E(<1,2,2,1,1>) = {5}
• E(<2,1,1,2,1>) = {2}
• E(<2,1,2,1,1>) = {4}
In the remaining 16 states, the set of winning strategies for debater 1 is empty.

As Glazer and Rubinstein note, this decision rule leads the listener to three mistakes: In the states <1,1,2,2,2> and <2,2,1,1,2>, debater 1 has a winning strategy (cite witness 1 and 3, respectively) even though a majority of the witnesses support debater 2's case; in the state <1,2,1,2,1>, he has no winning strategy even though a majority of the witness support his case.