PROBLEM SET #1 – DECISION PRINCIPLES
This problem set focuses on payoff matrices for (one-player) games against nature. Remember the basic set-up. The player chooses a strategy (normally a row in the matrix). Nature chooses a contingency (normally a column of the matrix). The player has complete information concerning the payoff matrix but must choose his strategy without knowing what contingency nature will choose. The player’s chosen strategy and nature’s chosen contingency define the outcome of the game, i.e., the cell in the matrix at the intersection of the chosen row and column. The numbers in the cells give the payoffs to the player — i.e., they indicate what outcomes are “good” and what are “bad” (and in what degree) for the player. The player aims to get the largest payoff possible. But nature is indifferent over all outcomes and gets no payoff (which is why there is only one number in each cell).
Here is a quick review of decision principles the player might follow (assuming that the player chooses rows).
Maximax principle (“aim for the best”):
(1) find the maximum payoff in each row, and
(2) choose the row with the maximum of the maximums.
Does a player always have a maximax strategy? Yes No
Is it always unique? Yes No
Maximin Principle (“avoid the worst’):
(1) find the minimum payoff in each row (its security level), and then
(2) chose the row with the maximum of the minimums (the highest security level).
Does a player always have a maximin strategy? Yes No
Is it always unique? Yes No
Maximize Average Payoff (“don’t focus on the best outcome or the worst outcome but on the average outcome”):
(1) add up all the payoffs in the row,
(2) divide by the number of contingencies, and
(3) choose the row with the highest average.
Does a player always have a average payoff maximizing strategy? Yes No
Is it always unique? Yes No
Maximize Expected Payoff (i.e., take account of the differing probabilities of contingencies):
(1) determine (or make a subjective estimate of) the probability pk of each contingency ck (where pk > 0 and ∑p = 1),
(2) multiply each payoff by its probability and add these products together to get the expected payoff of each row, and
(3) choose the row with the highest expected payoff.
Note: average and expected payoff are the same in the event each contingency has equal probability.
Does a player always have an expected payoff maximizing strategy? Yes No
Is it always unique? Yes No
Dominance (or “Sure Thing”) Principle:
Definition: strategy sk dominates strategy sh if
(1) sk gives at least as high payoff as sh in every contingency and a higher payoff in at least one contingency.
Basic Principle: don’t choose a dominated strategy (or, always choose an undominated strategy).
Does a player always have an undominated strategy? Yes No
Is it always unique? Yes No
Definition: a strategy is dominant if it dominates every other strategy.
Corollary Principle: always choose a dominant strategy, if you have one.
Does a player always have a dominant strategy? Yes No
Is it always unique? Yes No
I. Answer the following questions pertaining to each of the three payoff matrices.
|
|
c1 |
c2 |
|
s1 |
4 |
5 |
|
s2 |
2 |
8 |
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) s1 maximizes expected utility
(b) s2 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
II.
|
|
c1 |
c2 |
c3 |
|
s1 |
3 |
5 |
7 |
|
s2 |
4 |
2 |
10 |
|
s3 |
5 |
5 |
5 |
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) s1 maximizes expected utility
(b) s2 maximizes expected utility
(c) s2 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
III.
|
|
c1 |
c2 |
c3 |
c4 |
|
s1 |
4 |
8 |
3 |
2 |
|
s2 |
6 |
3 |
4 |
2 |
|
s3 |
3 |
5 |
3 |
2 |
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) s1 maximizes expected utility
(b) s2 maximizes expected utility
(b) s2 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
Now “transpose” each matrix, i.e., suppose the player chooses columns and nature chooses rows and answer the same set of questions for each.
Ia.
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) c1 maximizes expected utility
(b) c2 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
IIa.
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) c1 maximizes expected utility
(b) c2 maximizes expected utility
(c) c2 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
IIIa.
1. What is the player’s maximax strategy?
2. What is the player’s maximin strategy?
3 What strategy maximizes the player’s average payoff?
4. Can you find a probability distribution of contingencies such that
(a) c1 maximizes expected utility
(b) c2 maximizes expected utility
(c) c3 maximizes expected utility
(c) c4 maximizes expected utility
4. Does the player have a dominated strategy?
5. Does the player have a dominant strategy?
After the player has chosen his strategy and discovers what contingency nature has chosen, the player may regret his strategy choice and wish he had chosen a different strategy. Can you identify the condition under which a player would never have reason to regret his strategy choice?
Which decision principles do you think are most justifiable? Least justifiable?
Preview to think about only (no written answer required). Suppose that “nature” is no longer a disinterested player but rather nature is “malevolent” and is trying to hold the player’s payoff down to a minimum. Under this circumstance, do some decision principles become more justifiable? Less justifiable?