Thursday, January 28, 2010

The Opinion game - variations on a voting game

In my own personal throwback to the 1970's, when these kind of things were all the rage, I have thought up a game, with a few variations, that I believe can be of use in the modeling of opinions formed by groups.

This game is played with one player. There is a stack of identically backed tiles, each having a color on the reverse side, one of m possible colors. In each round, a new tile is revealed, and the player must pick a color.

They are awarded points on the following basis:
+1 point if they pick a color that is shown on the greatest number of tiles revealed thus far
-4 points if they changed their color from last round (no penalty for this in the first round)

Naturally, the behavior of the player will depend on the distribution of tiles. If the tiles are chosen ahead of time from an equally distributed lot, they will behave differently than if each tile's color is an independent random event.

Because the analysis is easier when each tile is independent, I will focus on this case. If there are 2 possible colors - say red and blue - the minimum number of rounds that must pass before a rational player has an incentive to change his or her opinion is 8. This follows from the following argument and example (WOLOG):
  1. We may limit ourselves to the simple case where one red tile is revealed then all blue tiles after that.
  2. It never makes sense to change one's mind within four rounds of the last round
  3. If red is behind by p tiles, the total expected payout of changing is derivable from the binomial distribution as follows: let n be the number of rounds remaining; the payout is a function of the path traveled to the final distribution. There are 2^n paths; we care about the difference between this outcome and the continuation of the red choice, thus equality of red and blue tiles is worth 0. Each path decomposes into previously calculated paths.
The shortest game in which switching makes sense is as follows:
Round 1:
Red tile revealed
Pick red

Round 2:
Blue tile revealed
Pick red (tied numbers; the "change cost" can be avoided)

Round 3:
Blue tile revealed
Pick red (at this point nine more rounds must be played for the expected value of changing to exceed 0)

Round 4:
Blue tile revealed
Pick blue (at this point, only 4 more rounds must be played for this choice to pay off)

Rounds 5 - 8:
regardless of tile revealed, player picks blue (see assertion 2)

Conclusions -

Studies of communication tend to emphasize the "deeply entrenched" nature of our opinions. This game confirms that if there is a social cost to changing an individual's opinion (such as alienation from your like-minded bretheren), a very large preponderance of evidence may be necessary to counterbalance this loss. Considering our number of choices in daily life, and the marginal benefits (if any) that abstract knowledge poses for the individual, it is no wonder that people tend to believe crazy things.

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