MAT109 · Erich Prisner · Franklin College · 2007-2011

# Mini Le Her

Note to the Teacher: ..........

Prerequisites: ....

MINI LE HER 2(q,k): This game is played with a q+k cards deck, q Queens and k Kings. Ann and Beth get both randomly a card at which they look without showing it to their opponent. A third card is put, back up, on the desk. Now Ann can decide whether she wants to change cards with Beth (of course without knowing what card Black has). Next Beth has the opportunity to exchange her card with that lying on the desk. Now both players reveal their cards, the one with the higher one wins 1 unit from the other. In case of a tie nobody gets any payoff.
The extensive form of the case q=k=4 is shown below:

Ann has two information sets, holding a Q or a K, with two moves each, therefore she has four pure strategies "EE", "ED", DE", and "DD", where "E" stands for exchange and "D" ("don't") for no exchange. Beth has six information sets: Holding a "Q" with Ann having a "Q", or holding a "Q" with Ann holding a "K", or holding a "K", with Ann holding a "Q", or holding a "K" with Ann holding a "K", or holding a "Q" and not knowing what Ann has (since she didn't exchange), or holding a "K" with not knowing what Ann holds (since Ann didn't exchange). In each of these six information sets she has two options, "E" and "D", therefore Beth has 26=64 pure strategies. Therefore the normal form of the game is a 4 × 64 matrix.
However, the game has four perfect-information-subgames which can be analyzed easily using Backwards Induction. They start at the four positions where Beth has to move and knows both of their cards since Ann has exchanged. In each of these cases it is clear what Beth has to do. We cut these subgames, and put the expected payoffs in to get the following truncated extensive form:

Here Beth has only two information sets left, and four pure strategies: "EE", "ED", "DE", and "DD". We get the following 4 × 4 matrix as normal form:
 EE ED DE DD EE -2/7 -2/7 -2/7 -2/7 ED 2/7 1/7 3/7 2/7 DE -4/7 -5/7 -3/7 -4/7 DD 0 -2/7 2/7 0

MINI LE HER 3(j,q,k): This game is played with a j+q+k cards deck, j Jacks, q Queens and k Kings. Ann and Beth get both randomly a card at which they look without showing it to their opponent. A third card is put, back up, on the desk. Now Ann can decide whether she wants to change cards with Beth (of course without knowing what card Black has). Next Beth has the opportunity to exchange her card with that lying on the desk. Now both players reveal their cards, the one with the higher one wins 1 unit from the other. In case of a tie nobody gets any payoff.
................... Here is the truncated extensive form of MINI LE HER 3(4,4,4):

### References

• [FairVote] Who picks the president? A report by FairVote, http://archive.fairvote.org/media/research/who_picks_president.pdf

### Exercises

1. Use the data in this Excel sheet to graph the money spent on ads for both candidates versus the number of electoral votes, for all swing states. Are these numbers again roughly proportional? Can the pattern that those candidates win who put in less effort into the state be confirmed for this data?
2. Who has an advantage in ELECTION(7,8,13|-1,-1,2|5,5)?
3. Analyze ELECTION(7,8,13|-1,2,-1|3,3).
4. What happens in ELECTION(7,8,13|-1,3-1|3,3)? How many resources do the players send into the districts on average? How do things change in ELECTION(7,8,13|-1,3-1|4,4)? What happens in ELECTION(7,8,13|-1,3-1|5,5)?