Tuesday, April 2, 2013

Dilemma 2: Dilemma Tournaments

Dilemma Tournaments

One route to building cooperation is by repeated play; this makes reciprocation and negotiation possible. Given a tournament of dilemma games, here’s how to assign an outcome to the tournament:

Sum up the payoffs of play; compare with pre-arranged goal.
If both players meet goal, then             Truce
If only one player meets goal, then      Win/Lose
If neither meet goal, then                     Draw

Meets Goal       Doesn’t
A:         Meets Goal                   Truce                Win/Lose
Doesn’t                         Lose/Win           Draw

This method averages a run of games. If on average we truce, then we truce the tournament; ditto for win, lose, and draw.      
Call win and truce the “upper” outcomes, or “prosperity”; draw and lose the “lower” outcomes, or “poverty”. This method awards upper outcomes to players who meet the goal.
The goal chosen depends on the scoring system. For the scoring system Lose = 0, Draw = 1, Truce = 2, and Win = 3, let Goal  =  (3/2)*n, where n is the number of plays. I call this the “Army” scoring system; the goal is to cover enough ground.

For the scoring system Lose = -3, Draw = -1, Truce = 1, and Win = 3, let Goal = 0. I call this the “Navy” scoring system; the goal is to stay afloat!
In general, let Goal =  (v)*(number of plays) , where v = “average outcome” =  (W+L+D+T)/4. I call this the “Charlie Brown” system; the goal is to be better than mediocre.

It turns out that, for best results, the tournament should end unexpectedly. When it ends should not be too well known at the start of play. (Read “The Unexpected Departure” below to see why.)
To ensure this, we will use an “open bounding” system for playing dilemma tournaments. Under open bounding, we use a randomizing device (coin, card, dice, etc.) at the end of each round to determine if there will be a replay. That way there will eventually be a last round, but we will never be sure when that last round will be.
The “Shadow of the Future” is the expected number of plays in the tournament. If the tournament is openly bounded with replay probability w, then the shadow of the future equals:
SF   = 1 + w + w2 + w3 + ...   =    1  /  ( 1 - w ).
For instance, if we flip a coin each turn, then w = 1/2 and SF = 2 turns. If we roll a die and replay if the roll is at least 3, then w = 2/3 and SF = 3. If we draw a card and replay unless wrong suit, then w = 3/4 and SF = 4. If we roll two dice and replay if the roll is at least 4, then w = 11/12 and SF = 12.

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