Probability and Expected Utility Economics 282 - Introduction to Game Theory Shih En Lu Simon Fraser University ECON 282 (SFU) Probability and Expected Utility 1 / 12
Topics 1 Basic Probability 2 Preferences over Lotteries and Expected Utility ECON 282 (SFU) Probability and Expected Utility 2 / 12
Notation and Basic Rules Suppose A and B are events (e.g. "it rains tomorrow in Vancouver," "the coin flip comes up heads," etc.). We use the following notation: P(A) is the probability of event A occuring P(AB) (or P(A B)) is the probability of events A and B both occurring; P(A B) is the probability of either event A or event B occurring; P(A B) is the probability of event A conditional on event B occurring. Law of Conditional Probability: P(A B) = P (AB ) P (B ) Inclusion-Exclusion Principle: P(A B) = P(A) + P(B) P(A B) If A and B are mutually exclusive (cannot both occur), then P(A B) = 0, and therefore P(A B) = P(A) + P(B). ECON 282 (SFU) Probability and Expected Utility 3 / 12
Independence Definition: Events A and B are independent if P(AB) = P(A)P(B). By the Law of Conditional Probability, this is equivalent to saying P(A B) = P(A). Interpretation: Learn whether one of the events will occur does not change how likely you believe the other event is. Example: "The coin flip comes up heads " and "it rains tomorrow in Vancouver" are independent events. Example: "It rains tomorrow in Burnaby" and "it rains tomorrow in Vancouver" are not independent events, i.e. they are correlated. ECON 282 (SFU) Probability and Expected Utility 4 / 12
Choice Under Uncertainty We have learned about preferences and utility functions over sure outcomes. But life is full of uncertainty! You often have to decide between choices that each lead to an uncertain outcome. Today s goal: represent preferences over uncertain outcomes. This is important for game theory: how do you best-respond to others when they use a non-pure strategy? ECON 282 (SFU) Probability and Expected Utility 5 / 12
Example of Choice Under Uncertainty You don t know if it s going to rain, and you have to decide whether to carry an umbrella. If you carry an umbrella: 10% chance you lose it, 60% chance you carry it around needlessly, 30% chance you use it If you leave umbrella at home: 0.1% chance you lose it, 66.6% chance you don t need it, 33.3% chance you get wet What factors matter in your decision? Your decision depends on the probability of each outcome, and on how much you like/hate each outcome. We will assume that these are the only relevant factors (and ignore things like the "Ellsberg paradox"). ECON 282 (SFU) Probability and Expected Utility 6 / 12
Lotteries: Notation and Terminology Suppose a situation has n possible and mutually exclusive outcomes, labeled 1, 2,..., n. Example: Lose umbrella (1), carry it needlessly (2), use umbrella (3), don t carry and don t need it (4), get wet (5) A lottery [p 1, p 2,..., p n ] is a list of probabilities, where p i is the probability that outcome i occurs. Example: Carrying the umbrella leads to lottery [0.1, 0.6, 0.3, 0, 0], not carrying it leads to lottery [0.001, 0, 0, 0.666, 0.333]. Note: because p 1, p 2,..., p n are the probabilities of all possible and mutually exclusive outcomes, we must have p 1 + p 2 +... + p n = 1. ECON 282 (SFU) Probability and Expected Utility 7 / 12
Expected Utility Suppose outcome 1 gives you utility u 1, outcome 2 u 2, and so on. What is your utility from lottery L = [p 1, p 2,..., p n ]? Natural answer: p 1 u 1 + p 2 u 2 +... + p n u n, which is the L s (von Neumann-Morgenstern) expected utility. But even if the u i s represent your preferences over outcomes, expected utility may not represent your preferences over lotteries. Example: Fido prefers chicken (outcome 1) over pears (outcome 2) over apples (outcome 3). Assigning u 1 = 2, u 2 = 1 and u 3 = 0 would represent its preferences over these sure outcomes. But suppose Fido prefers [0.4, 0, 0.6] over [0, 1, 0]. With the above utilities, does expected utility represent Fido s preferences over lotteries? So it s important to assign the right utility to each outcome - not just the order matters (ordinal utility), but size matters too (cardinal utility). ECON 282 (SFU) Probability and Expected Utility 8 / 12
Axioms for Expected Utility (I) Given preferences over lotteries, it s not always possible to find utilities over outcomes u 1, u 2,..., u n such that expected utility represents the said preferences over lotteries. Just as you needed assumptions on preferences over outcomes to build a utility function representing them, you need assumptions on preferences over lotteries to build an expected utility function representing them. There are four required axioms. The first two are the same as the axioms needed on preferences over outcomes, but now applied to lotteries: 1 Completeness: For any lotteries L and L, either L L, L L, or L L. 2 Transitivity: If L L and L L, then L L. ECON 282 (SFU) Probability and Expected Utility 9 / 12
Axioms for Expected Utility (II) The other two axioms are more complicated. Here are verbal (and therefore non-rigorous) descriptions for them: 3. Continuity: Given a best, a medium and a worst lottery, respectively L, L and L, there is a mixture of L and L (i.e. "with some probability, you get L; otherwise, you get L ") that s exactly as good as L. This seems like a sensible axiom, but its applicability is in doubt in some extreme cases. Suppose L is winning $100.01, L is winning $100, and L is dying... Most microeconomic theorists find the last axiom most problematic: 4. Independence: If the agent prefers lottery L to lottery L, then the agent will also prefer a mixture of lotteries L and L to a mixture of lotteries L and L if both mixtures place the same probability on L. In other words, your preference over two lotteries isn t affected by mixing in a third. This assumes away things like the "Allais paradox." ECON 282 (SFU) Probability and Expected Utility 10 / 12
Expected Utility Theorem Theorem If preferences satisfy Axioms 1-4, then it is possible to assign a real number (utility) u i to each outcome i = 1, 2,..., n such that L L if and only if U(L) U(L ), where U([p 1, p 2,..., p n ]) = p 1 u 1 + p 2 u 2 +... + p n u n. Due to John von Neumann and Oskar Morgenstern. The theorem tells us that these u i s exist, but it doesn t tell us what they are. The proof of the theorem is beyond the scope of this course... ECON 282 (SFU) Probability and Expected Utility 11 / 12
Take Away Message Almost all of game theory assumes that the four axioms hold so that agents preferences admit an expected utility representation. We will do so throughout the rest of this course. This is an intuitive assumption, but there are cases where it does not match the real world. The utilities assigned to outcomes in the description of a game are chosen such that each player s preferences over lotteries are represented by her expected utility function. Implication: We will be relying on cardinal utility. (And in fact already relied on cardinal utility when studying repeated games.) ECON 282 (SFU) Probability and Expected Utility 12 / 12