Imperfect Information

Imperfect Information

by Munazza, Maureen, and Yixue

In her computer science course at Mount Holyoke, Ashley was instructed to program a game. She designed a simple dice game similar to “Pig” with two players. Two dice were pictured on the screen, and the first player rolled the dice until he/she rolled a one, at which point all of the accumulated points would disappear, and the second player would begin rolling. The only way to hold points past turns was to click a hold button at the bottom of the screen, at which point it would become the other player’s turn. The first player to accumulate 100 points would win.

One night, while playing this game in the Pearsons common room, Ashley and her friend Cam began discussing rationality. “After all,” Cam argued, “there must be an optimal way to play this game so that I have the best chance of winning possible.”

“True,” Ashley countered, “but it’s so tempting to roll just one more time, and then again, and again… for example, I’ve rolled more than five times without rolling a one and have 41 points.” Intrigued by the conversation, Cam began observing other friends who played the game, and made some interesting discoveries about rationality and information.

If an individual is rational, then they should make decisions that maximize their personal utility based on all available information, and when discussing consumer choices in class, we generally assume perfect information, which leads to perfectly rational decisions. However, in reality, no one has access to perfect information, and this affects choices and behavior. When playing Pig, each player should choose the strategy that is most efficient in order to gain his/her desired outcome by rationally considering all options and then playing accordingly, without being tempted by the lure of trying their luck on a less optimal but “flashier” strategy. 

After watching many friends play this game Cam realized there were several factors that, on the surface, made some player decisions seem irrational. This was because different players had different goals. Some players were most interested in winning against their opponent, while others were most interested in gaining the highest number of points personally, irrespective of what their opponent gained. The third category of players were most interested in gaining the highest score on each roll, irrespective of their final scores and their opponent’s final score.

These three categories of players made very different decisions. Those who wanted to win against their opponent almost always pressed hold after one or two rolls, capturing small amounts of points on almost every turn. The second category rolled three or four times before pressing hold and sometimes ended up with more points than the first group, but not necessarily winning against their opponent. The third category rolled as many times as they could, often losing points after rolling a one, but occasionally gaining really large amounts of points per turn, which was celebrated as a victory in and of itself. Therefore, each player was making a rational decision depending on his/her personal goals while playing the game.

Cam’s second observation was based on information. Assuming that each player did not have perfect information about the specific mathematical probabilities of the game or about their opponent’s playing strategies, she noticed that some players modified their strategies as they gained more information by observing their opponent. Therefore, as information increased, each player adjusted his/her strategy accordingly.