Let's play the dollar auction. It's like any other, with the winning bid handing over the bid and collecting the thing to be auctioned. But there is this difference: the runner-up also hands over its bid, but gets nothing in return. I am auctioning a dollar. Go!
The reader might think this is a simple game and that I, as the auctioneer, will be losing some money shortly. According to author Mérő the average winning bid is $3.40! I intend to try it at my first large gathering (one needs a critical mass of potential bidders for it to be effective).
This was such a fun book, but also a difficult read, requiring two passes for me to write intelligently about it.
There are four basic games, which I list below in order of my familiarity with them.
Chicken. One variation has two cars speeding towards a cliff, the first car to turn away loses. Another variation has two cars speeding towards each other, again the first car to turn away loses. Here, close cooperation yields the best result, if both players agree to turn off at the same time. Strong competition yields the worst result. The best strategy, given the assumption that the players are not allowed to scheme together, is to appear as irrational and suicidal as possible.
Prisoner's dilemma. Two players together commit and crime and speed away in separate cars. Both are stopped for speeding and subsequently arrested for the crime. Neither has positive or negative feelings for the other, the partnership designed to last only for the length of the crime. The prosecutor approaches each, separately, the following proposition.
I do not have a strong case for the crime unless one or both of you confesses, though I have an easy case on the speeding. If you confess to the crime and implicate your partner I will set you free and he will receive a ten year jail sentence. If he confesses and you do not you will be sentenced to ten years and he will walk free. If you confess and he does also then your testimony will be less valuable to me and I will not set either of you free and will instead seek (and receive) a five year sentence. If neither of you confesses then I have no evidence on the crime and each of you will receive a one-year sentence for the speeding.
The possible outcomes are:
- The players cooperate with each other by confessing, each spends five years in jail.
- The players compete with each other by refusing to confess, each spends one year in jail.
- One player cooperates and the other competes; the cooperating player spends 10 years in the jail and the other walks free.
In this game mutual competition yields the best result.
Gift of the Magi. A couple is planning their evening. They lay out their preferences but then need to leave for work. They did not reach agreement, will have no opportunity to talk during the day, and must leave immediately after work to reach either option in time. The options are these. The husband prefers a sporting event, the wife a concert. Both prefer most, however, spending the time together in either activity. If each acts competitively and pursues their preferred activity they will not achieve the optimal outcome, spending time together; they will, however, somewhat enjoy the evening. If each acts cooperatively and pursues the other's preferred activity they will again not achieve the optimal outcome and both will attend events neither enjoys. The best outcome is achieved if one party acts competitively and the other cooperatively.
Holding the door. Two people approach door. They both most value passing through and they both also value holding the door for the other, allowing the other to pass first. If each acts competitively by refusing to pass through the door before the other they achieve the worst outcome. If each acts cooperatively then they achieve the best outcome. If one acts competitively and one cooperatively then the competitor gets the best outcome and the cooperator gets an outcome with some positive value.
Returning to the dollar auction, a few months after writing the above I had a chance to play that game and another, the lowest-number game. The lowest-number game is played as follows: everyone writes on a piece of paper their name and a positive integer. The player writing the lowest integer wins, with the exception that if two or more players select the same number then all such players are disqualified.
In the dollar auction the best I could do, with an audience of over 100, was to get bids of $1.01 and $1.00. Perhaps my audience was especially intelligent. I also played the lowest-number game with this audience. The winning answer was 5. Several players chose 1. Sevens, and numbers ending in 7, were popular (7, 17, 27, 37, 1337, 11337, for example). One player wrote 1.3 billion.
I really enjoy the lowest-number game because it can be played by anyone old enough to write his or her name. Wisdom and experience do not seem to provide an edge, based on my observations of my children and their friends. And, interesting patterns emerge when repeating play amongst the same players. One eight-year-old consistently wrote "1" on her paper. If you are competing against her, what is the best strategy? Also write "1" preventing her from winning but also eliminating your own chances? Never write "1", risking the situation where the other players follow suit? Or, as the author states, employing a mixed strategy. I suppose a mixed strategy in this case is writing "1" sometimes but not others and thus encouraging this player to select a different number
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