Utility Theory and Game Theory (5/1/02)
Expected utility theory - decision theory for a single agent
Example 1: Planning a party - a game against nature
Our agent is planning a party, and is worried about whether it will
rain or not. The utilities and probabilities for each state and action
can be represented as follows:
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Nature's states:
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(p=1/3) |
(~p=2/3) |
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| Party planner's possible actions: | Outside |
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| Inside |
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The expected utility of an action A given uncertainty about a state
S = Probability(S|A)*Utility(S|A) + Probability(not S|A)Utility(not S|A)
Note that action A can be viewed as a compound gamble or outcome.
Also, note that the probability of a state can depend on the agent's
choice of action, although, in the above example, it does not.
For the party problem:
EU(Outside) = (1/3)(1) + (2/3)(3) = 2.33;
EU(Inside) = (1/3)(2) + (2/3)(2) = 2
Therefore, choose Outside, the action with the higher expected utility.
Summary of the formal theory of expected utility
Definitions.
1. S is a set of outcomes {x,y,z,w}
2. x ] y means "x is preferred to y" (also known as strict preference)
3. x ~ y means "x is viewed indifferently relative to y"
4. x ]~ y means "x is either preferred or viewed indifferently relative
to y" (also known as weak preference)
5. (x,p,y) means a gamble (an uncertain outcome, or a lottery) in which
outcome x will be received with probability p, and outcome y will be received
with probability 1-p.
Axioms. For all x,y,z,w in S, and p,q in (0,1):
1. Closure. (x,p,y) is in S.
2. Weak ordering.
(a) x ]~ x. (Reflexivity)
(b) x ]~ y or y ]~ x. (Connectivity)
(c) x ]~ y and y ]~ z imply x ]~ z. (Transitivity)
3. Reducibility. ((x,p,y),q,y) ~ (x,pq,y).
4. Independence. If (x,p,z) ~ (y,p,z), then (x,p,w) ~ (y,p,w).
5. Betweenness. If x ] y, then x ] (x,p,y) ] y.
6. Solvability. If x ] y ] z, then there exists p such that y
~ (x,p,z).
Theorem: (J. von Neumann & O. Morgenstern, Theory of Games and Economic
Behavior, 1944)
If axioms 1-6 are satisfied for all outcomes in S, then there exists
a real-valued utility function u defined on S, such that (1) x ] y if and
only if u(x) > u(y), and x ~ y if and only if u(x) = u(y); (2) u(x,p,y)
= pu(x)+(1-p)u(y); (3) u is an interval scale, that is, if v is any other
function satisfying 1 and 2, then there exist real numbers b, and
a>0,
such that v(x) = au(x)+b.
Problems with the theory of expected utility
(1) Human preferences do not obey the assumptions of the theory (e.g. Quattone and Tversky, 1988)
(a) Violations of axioms (transitivity, reducibility, independence)
(b) Violations of invariance (framing effects: reference point dependency and loss aversion, ratio-difference principle)
(2) Assumes there are no rational "opponents" or other intelligent agents
who are part of the game.
(Noncooperative) game theory - decision theory for more than one agent, each acting autonomously (no binding agreements)
In the examples below, we'll assume two self-utility maximizing agents (or players), each of whom has complete information about the options available to themselves and the other player as well as their own and the other's payoffs (utilities) under each option.
Example 2 - Friends hoping to see each other
Consider two people, Chris and Kim. They both enjoy each other's
company, but neither can communicate with the other before deciding whether
to stay at home (where they would not see each other) or go to the beach
this afternoon (where they could see each other). Each prefers
going to the beach to being at home, and prefers being with the other person
rather than being apart. This game can be represented by the following
normal (or matrix) form:
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Kim
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| Chris | Home |
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| Beach |
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Each player has a set of strategies (={Home,Beach} for both players
in this example).
Specifying one strategy i for the row player (Chris) and one strategy
j for the column player (Kim) yields an outcome, which is represented as
a pair of payoffs (Rij,Cij), where Rij is the utility the row player receives,
and Cij is the utility the column player receives.
In this example, going to the beach is a (strictly) dominant strategy for each player, because it always yields the best outcome, no matter what the other player does. Thus, if the players are both maximizing their individual expected utilities, each will go to the beach. So Beach-Beach is a dominant strategy equilibrium for this game. Because of this, Kim and Chris, if they are rational, do not need to cooperate (make an agreement) ahead of time. Each can just pursue their own interest, and the best outcome will occur for both.
Example 3 - "Friends" with asymmetric preferences
Now consider Betty and John. John likes Betty, but Betty doesn't
like John that much. Each knows this, and neither wants to call the
other before deciding what to do this afternoon: stay at their respective
homes or go to the neighborhood swimming pool. Here is the normal
form:
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John
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| Betty | Home |
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| Pool |
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In this case, Betty's best strategy depends on what John does. But if she assumes John is rational, she will reason that he will not stay home, because going to the pool is a dominant strategy for him. Knowing this, she can decide to stay home (because 2>1). This is called iterated dominance. In this example, Betty gets higher utility than John because of their relative preferences, and John gets less utility than he would have if Betty wanted to be with him.
In this example, Pool-Home (3,0), Home-Pool (2,1), and Pool-Pool (1,2) are all Pareto optimal outcomes. An outcome is Pareto optimal (or efficient) if no agent can be made better off than that outcome without making another agent worse off. The equilibrium outcomes in both this example and the previous one are Pareto optimal.
Example 4 - Prisoners' dilemma
Consider Stan and Leland, two prisoners who have each been offered a
deal to turn state's witness (defect) against the other. They can't
communicate. They had orginally agreed to remain in solidarity,
i.e. not testify against each other, but since the agreement cannot be
enforced, each must choose whether to honor it. If both remain in
solidarity, then they will each only be convicted of a minor chage.
If only one defects, then the state will throw the book at the other and
let the defector go. If they both defect, each will get convicted
of a serious charge. The payoff matrix (higher positive utility implies
a better outcome) is as follows:
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Leland
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| Stan | Solidarity |
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| Defection |
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In this game, the strategy of defection is weakly dominant for each player, meaning that whatever the other player does, defecting yields an outcome at least as good and possibly better than remaining in solidarity would. Note that if the bottom right cell payoffs were (2,2) instead of (1,1), then defecting would be strictly dominant for each player. Either way, Defection-Defection is a dominant strategy equilibrium. However, it is not Pareto optimal. Both players could be made better off if neither defected against the other.
This is an example of a social dilemma: a situation in which each agent's autonomous maximization of self-utility leads to an inefficient outcome. Such a situation can occur for any number of people, not just two. An agreement by two people to trade with each other (involving goods, services, and/or money) set's up a prisoners' dilemma-type game whenever the agreement cannot be enforced.
Example 5 - Coordination
Let's go back to Chris and Kim. They are going to the same conference,
and each is expecting the other to be there, but they haven't seen each
other yet. The conferees have their choice of two activities on the
first afternoon: swimming or hiking. They both hope to see each other
-- if they don't they will have no fun, and each prefers swimming over
hiking. They must each decide what to do before knowing where the
other is going. Here is the normal form:
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Kim
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| Chris | Swim |
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| Hike |
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The best outcome is obviously Swim-Swim, but going swimming is not dominant for either player. Both Swim-Swim and Hike-Hike have the property that each player's strategy is the best (or tied for the best) response to the other player's strategy in that pairing. This defines a more general equilibrium notion called the Nash equilibrium. The dominance equilibria of examples 1-3 are all Nash equilibria as well.
A third equilibrium exists in this game involving what are called mixed strategies. A mixed strategy is a probability distribution over the pure strategies (which are Swim and Hike for each player in this example). (Note that the players do not have to have the same set of strategies available to them, even though that has been the case in all our examples.) In this example, if each player individually throws a die and goes swimming if the die comes up 1 or 2, and goes hiking if the die comes up 3, 4, 5, or 6, the resulting expected utility (2/3 for each player) cannot be improved upon for either player given that the other player uses this strategy.
In 1950, John Nash (depicted somewhat fictitiously in the film A Beautiful Mind -- the book is more accurate!) proved that every finite game, involving any number of players, has at least one (Nash) equilibrium, though there might not be any that involve only pure strategies for all players. In this example, there are three equilibria: the mixed strategy equilibrium (Swim,1/3; Hike,2/3)-(Swim,1/3;Hike,2/3), and two pure strategy equilibria -- Swim-Swim and Hike-Hike.
When there is more than one equilibrium, and players cannot make binding agreements, they must try to coordinate to arrive at an equilibrium outcome. When only one equilibrium is also Pareto optimal, as Swim-Swim is in this case, that fact should suggest to rational players that it will be the one around which they coordinate. Many other criteria for equilibrium selection have been studied (e.g. focal points, subgame perfection, stability -- see the reading on game theory).
Example 6 - "Battle of the sexes"
Finally, let's consider Roy and Jen. They are going to the same
conference as Kim and Chris in example 5. They each would prefer
to be in the same place (the swim or the hike), but their preferences differ
about which it should be. Roy would rather go swimming, and Jen would
rather go hiking. Here is the matrix form:
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Jen
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| Roy | Swim |
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| Hike |
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This game has three Nash equilibria: Swim-Swim, Hike-Hike, and (Swim,2/3;Hike,1/3)-(Swim,1/3;Hike,2/3). Note that the mixed strategies differ for each player in the third equilibrium: each goes to their preferred activity with 2/3 probability.
All of the equilibria are Pareto optimal this time, so that does not help for selection. Only the mixed strategy equilibrium results in equal expected utilities for the two players, so if both value equality or symmetry, this might be the focal point. But of course it will be difficult for Roy and Jen to see that unless they have studied game theory!
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