Part One: Probability

Ellsberg's Urns₁

A Pair of Urns

We have two (2) covered bowls (also called closed urns), each containing a mix of blue and green chips. The chips are identical except for color.

The first bowl contains equal proportions (or fractions) of each type of chip. Each blue chip in the first bowl has precisely one green twin, so that exactly 50 of the chips in the first bowl are blue chips, and all of the remaining chips in the first bowl are green chips.

All that we know of the second bowl is that it contains only yellow and red chips - each chip contained in the second bowl is either yellow or red . We do not know the proportions (or fractions) of each color (yellow or red ) of the chips contained in the second bowl.

The First Game

Suppose that we have a (fictitious) betting game based only on draws with replacement from the first bowl₂. Suppose that the player wins by guessing correctly the color of the chip drawn from the first bowl.

What is your strategy for this game? What is your rule for guessing the color on each single draw? How often do you expect to win ? Is there a "best" strategy for playing the first game ?

The Second Game

Suppose that we have a (fictitious) betting game based only on draws with replacement from the second bowl. Suppose that the player wins by guessing correctly the color of the chip drawn from the second bowl.

What is your strategy for this game? What is your rule for guessing the color on each single draw? How often do you expect to win ? Is there a "best" strategy for playing the first game ?

The Third Game

Suppose that this game begins with the player choosing one of the bowls from which to draw chips with replacement. Next, the (fictitious) betting game is based only on draws with replacement from the bowl of choice. Suppose that the player wins only when a blue or red chip is drawn from the bowl of choice.

What is your strategy for this game? Which bowl do you choose? How often do you expect to win? Is there a "best" strategy for playing the third game ?

This style of thinking is a form of decision theory or game theory - one starts the game with certain assumptions, plays the game under those assumptions, and wins or loses each time. From time to time, the assumptions and strategy might be changed based on the prior win/loss record.

1. D. Ellsberg, "Risk, Ambiguity, and the Savage Axioms," Quarterly Journal of Economics 75 (1961): 643-69.

2. On each draw from the bowl, we draw a chip from the bowl, observe the color of the chip and then replace it. Each successive draw from the bowl is based on the same set of chips in the bowl.