Chapter 11 What is Utility?

• A way of representing preferences

• Utility is not money (but it is a useful analogy)

• Typical relationship between utility & money:

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MultiAgent Interactions

• Each agent has preferences. Each agent gets utility depending on their choices and the choices of the other

• We can write this set of two utilities as a payoff matrix as follows.

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PayOff Matrices- Business plan. Success depends on other’s choices.- For simplicity, consider two players.- I make a choice, but the consequences of that choice depends

on what you do.- Let c(a1,a2) denote the consequence that results when I (agent 1)

choose action a1 and you (agent 2) choose action a2- Let the utility of that consequence be u1[c(a1,a2)], we often

abbreviate this as u1(a1,a2). - There is a sound mathematical theory called Game Theory for

dealing with multi-agent choice problems when every agent knows its utilities and the utilities of all other agents.

- the field of game theory came into being with the 1944 book Theory of Games and Economic Behavior by John von Neumann and Oskar Morgenstern. 1930 some early work began.

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Normal Form• When outcomes depend only on a single choice by me

and a single choice by you, then the game is said to be in normal form. This terminology represents the idea that the game is in a normative form, meaning a canonical form.

• The payoff matrix expresses games in normal form, but the game tree expresses games in extensive form- representing turn taking. It will probably be helpful to associate the phrase extensive form with the notion of a tree, and the phrase normal form with the notion of a payoff matrix.

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Strategy• Game theorists can reason about strategies, based on

contingencies, rather than discrete actions.• In this context, a strategy is not an attitude like play aggressively or

defend the goal, but rather a complete expression of what to do in every contingency. As described by Poundstone, – [A strategy] is a complete description of a particular way

to play a game, no matter what the other player(s) does and no matter how long the game lasts . A strategy must prescribe actions so thoroughly that you never have to make a decision in following it.

• A strategy is defined as a choice made in response to the previous choices made in the game.

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Prisoners’ dilemma – damaged property

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Both get large fines

Ned is suspended

Kelly gets wrist slapped

Kelly is suspended

Ned gets wrist slapped

Both get minimal

fines

Confess

Confess

Don’t Confess

Don’t Confess

Ned

Kelly

Prisoners’ dilemma

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2,2 5,0

0,5 3,3

Confess

(defect)

Confess

(defect)

Don’t Confess

(cooperate)

Don’t Confess

(cooperate)

Ned

Kelly

Solution Concepts• Three different approaches:1.minimax, maximin2.Nash equilibria3.Pareto optimal

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Maximin: look at worst case scenario. Pick result that maximizes the worst case

Confess(defect)

Not confess(cooperative)

Results 2 or 5 0 or 3

Worst Case 2 0

Best Response

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Suppose for a moment that I am P1 and that I know what P2 will choose.  Given that I know P2's choice, I can search through all of my choices and find the option that is the best response to his choice.  For the prisoner's dilemma, if P2 chooses to confess then the option that maximizes my payoff is to confess.  Similarly, if P2 chooses to not confess then the option that maximizes my payoff is to confess. 

Best response – simply the best thing I can do GIVEN I know what you will do.

BUT, I don’t know what you will do. It has to do with regret.

This prisoners dilemma problem makes it easy – as I pick the same thing either way.

Prisoners’ dilemma – Best Response

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2,2 5,0

0,5 3,3

Confess

(defect)

Confess

(defect)

Don’t Confess

(cooperate)

Don’t Confess

(cooperate)

Ned

Kelly

Best Response

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Suppose that instead of knowing exactly which choices P2 was going to make, I know only that P2 will play confess, say, 40% of the time and not confess 60% of the time. 

I can find the expected utility for confessing (.4*(2)+.6*(5) = 3.8 and the expected utility for not confessing (.4*(0)+.6*(3) = 1.8), which tells me that my best response to P2's strategy is to confess. The notion of a best response is very useful to help understand other solution concepts from multi-agent choice. 

For example, the maximin solution can be viewed as the best-response (i.e., the maximal payoff) when I believe that the other player will always be able to choose the thing that hurts me the most.

Prisoners’ dilemma

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2,2 5,0

0,5 3,3

Confess

Confess

Don’t Confess

Don’t Confess

Ned

Kelly

Note that no matter what Ned does, Kelly is better off if she confesses than if she does not confess. So ‘confess’ is a dominant strategy from Kelly’s perspective. We can predict that she will always confess.

Nash Equilibrium• Suppose you decide to look at the best choice for both of you.

Clearly, ”not confess” is best for both. • Assuming both of you pick that “best for both solution”

– You observe that if you confess, your utility improves.– You observe that if the other player confesses, HIS utility

improves.

– You notice that if you both confess, neither player has the motivation to change his mind.

• A Nash equilibrium is a set of solutions where every player's choice is a best response to every other player's choice. In other words, the equilibrium solution is made up of a set of individual choices that make up a joint action from which neither benefits by deviating.

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Nash Equilibrium – (Beautiful Mind)

• In general, we will say that two strategies s1 and s2 are in Nash equilibrium if:

1. under the assumption that agent i plays s1, agent j can do no better than play s2; and

2. under the assumption that agent j plays s2, agent i can do no better than play s1.

• Neither agent has any incentive to deviate from a Nash equilibrium- it is stable

• Unfortunately:1. Not every interaction scenario has a Nash equilibrium2. Some interaction scenarios have more than one Nash equilibrium

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Criteria for evaluating systems• Social welfare: maxoutcome ∑i ui(outcome) where ui is the utility for player i.• Surplus: social welfare of outcome – social welfare of status quo

– Constant sum games have 0 surplus (as all options sum to same total). Markets are not constant sum

• Pareto efficiency: An outcome o is Pareto efficient if there exists no other outcome o’ such that some agent has higher utility in o’ than in o and no agent has lower utility– if we maximize social welfare we have pareto optimal, but not vice versa– Not a very useful way of selecting strategies

• Individual rationality: Agent will do what is best for him• Stability: No agents can increase their utility by changing their strategies (given

everyone else keeps the same strategy). If I knew what my opponent would do, would I still be satisfied with my decision?

• Symmetry: I get same utility as you if our roles were reversed.• No dictator: no agent is inherently preferred.

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Example: Prisoner’s Dilemma• Two people are arrested for a crime. If neither suspect confesses, both get

light sentence. If both confess, then they get sent to jail. If one confesses and the other does not, then the confessor gets no jail time and the other gets a heavy sentence.

• (Actual numbers vary in different versions of the problem, but relative values are the same)

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2,2 5,00,5 3,3

Confess

Confess

Don’tConfess

Dominant Strategy Equil not pareto optimal

Maximize social welfare

Don’tConfess

Pareto optimal

Pareto Optimal• Both maximin and Nash equilibrium solutions are pretty

pessimistic; they frame the problem as a competitive game between my interests and your interests.

• This competition is not healthy because they both produce

a solution (to prisoners dilemma) with both confessing, and both therefore receiving next to least favorite outcome of all possibilities (5,3,2,0).

• The idea behind Pareto optimality is that some joint solutions should obviously be avoided because they are bad for everyone.

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Stated as ordinal (position) rather than cardinal (how much) values…

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2,2 4,1

1,4 3,3

Defect Cooperate

Cooperate

Defect

The term pareto efficient…• The term pareto efficient is named after Vilfredo Pareto,

an Italian economist who used the concept in his studies of economic efficiency and income distribution.

• He is also the one credited with the 80/20 rule to describe the unequal distribution of wealth in his country, observing that twenty percent of the people owned eighty percent of the wealth.

• If an economic system is not Pareto efficient, then it is the case that some individual can be made better off without anyone being made worse off. It is commonly accepted that such inefficient outcomes are to be avoided, and therefore Pareto efficiency is an important criterion for evaluating economic systems and political policies.

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Strategic Dominance• One of my actions is better than my other

choices regardless of what my opponent picks.

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(Example is a non-symmetric game)

P1 is always better off by playing CIs there a dominant strategy for P2? (No, why?)A strategically dominant solution is a solution which is a best response for every possible choice made by the other players.

Satisficing Equilibrium• There are other, non-traditional solution concepts that are

relevant for multi-agent games. • One of these solution concepts is the notion of satisficing

equilibrium. • The word satisfice means to strive for something that is

sufficient. A satisficing equilibrium occurs when agents have arrived at choices such that the consequence produced by these choices yields utilities that all agents are satisfied with.

• It is an equilibrium because a satisficing agent is content with what they have. If all agents are content, no agent has an incentive to change its actions, which means that the solution is stable.

• In real world, may not know (or have resources to evaluate) all options, but can make a decision knowing that it is “good enough”

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At seats: Show in normal form- wrestling

• there is a widespread practice in high school wrestling where the participants intentionally lose unnaturally large amounts of weight so as to compete against lighter opponents.

• In doing so, the participants are clearly not at their top level of physical and athletic fitness and yet often end up competing against the same opponents anyway, who have also followed this practice (mutual defection).

• The result is a reduction in the level of competition. Yet if a participant maintains their natural weight (cooperating), they could compete against a stronger opponent who has lost considerable weight.

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Utility: inconvenience (0,-1) + competitive advantage (-3,3)

Losing to compete Compete at normal weight

Losing to compete -1, -1 2, -3

Compete at normal weight -3,2 0,0

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So what is best option?

Game of Chicken•Consider another type of encounter — the game of chicken:

•(Think of James Dean in Rebel without a Cause)•Difference from prisoner’s dilemma:

Mutually going straight is most feared outcome.(Whereas sucker’s payoff is most feared in prisoner’s dilemma.)

straight swerve

straight -10, -10 10, 5swerve 5, 10 7, 7

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Kelly

Ned

Game of Chicken

•Is there a dominant strategy?•Is there a pareto optimal (can’t do better without making someone worse)?•Is there a “Nash” equilibrium – knowing what my opponent is going to do, would I be happy with my decision?

straight swerve

straight -10, -10 10, 5swerve 5, 10 7, 7

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Try this one

•Is there a dominant strategy?•Is there a pareto optimal (can’t do better without making someone worse)?•Is there a “Nash” equilibrium – knowing what my opponent is going to do, would I be happy with my decision?

coop defect

coop 5,5 0,0defect 0,0 10,10

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And this one

•Is there a dominant strategy?•Is there a pareto optimal (can’t do better without making someone worse)?•Is there a “Nash” equilibrium – knowing what my opponent is going to do, would I be happy with my decision?

coop defect

coop 1,0 4,3defect 3,3 5,2

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And this one

•Is there a dominant strategy?•Is there a pareto optimal (can’t do better without making someone worse)?•Is there a “Nash” equilibrium – knowing what my opponent is going to do, would I be happy with my decision?

coop defect

coop 1,0 4,3defect 3,3 5,2

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Free Rider•described by Poundstone• It's late at night, and there's no one in the subway station. Why not just hop over the turnstiles and save yourself the fare? But remember, if everyone hopped the turnstiles, the subway system would go broke, and no one would be able to get anywhere.

What's the chance that your lost fare will bankrupt the subway system? Virtually zero. The trains run whether the cars are empty or full. In no way does an extra passenger increase the system's operating expenses.

But if everybody thinks this way

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Normal form game*(matching pennies)

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Agent 1

Agent 2

H

H

T

T

-1, 1

-1, 1

1, -1

1, -1

*aka strategic form, matrix form

ActionOutcome

Payoffs

Extensive form game(matching pennies)

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Player 1

Player 2

H

H H

T

TT

(-1,1) (-1,1)(1,-1) (1,-1)

Action

Terminal node (outcome)

Payoffs (player1,player 2)

Player 2 doesn’t know what has been playedso he doesn’t know which node he is at.

Strategies• Strategy:

– A strategy, sj, is a complete contingency plan; defines actions which agent j should take for all possible states of the world. In these simple games, the state is always “the beginning”.

• Strategy profile: s=(s1,…,sn) – what each agent did (assuming n players).

– s-i = (s1,…,si-1,si+1,…,sn) - what everyone else did

• Utility function: ui(s)– Note that the utility of an agent depends on the strategy profile,

not just its own strategy– We assume agents are expected utility maximizers

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Normal form game*(matching pennies)

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Agent 1

Agent 2

H

H

T

T

-1, 1

-1, 1

1, -1

1, -1

*aka strategic form, matrix form

Strategy for agent 1: HStrategy for agent 2: TStrategy profile(H,T)

U1((H,T))=1U2((H,T))=-1

Battle of the Sexes

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2,2 4,3

3,4 1,1

Shopping Fishing

Fishing

Shopping

Wife preference

Husband preference

Consider the famous game of Battle of the Sexes.  In the game, a husband and wife must independently decide on a date activity.  The husband would prefer one form of entertainment, say fishing, and the wife would prefer another form of entertainment, say shopping for clothes.  Although both have their most preferred activity, both prefer being together to being alone.

Maximin

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2 3

4 1

Worst Choice

Fishing

Shopping

Wife preference

Husband preference

Wife Looks at worst option for her.Pick solution with maximizes

Reactions• Maximin – both should be selfish (and do what they want).• Not a great solution as if either “defects” (from the selfish

choice) makes both happier. • Is better than worst case of (1,1)

• In fact, either consequence that results when one is selfish and the other unselfish dominates (in the Pareto-optimal sense) the maximin solution, and both of these consequences are in equilibrium (since neither player benefits by unilaterally changing his/her mind). Unfortunately, with no way to communicate the players are left with making independent choices to try and reach an equilibrium.

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Reaction?

• Could just randomly pick a strategy – Termed mixed strategy.

How do you think that would work?

Battle of the sexes (cont)• Additionally, using a mixed strategy (picking each

option a fraction of the time) does not help their chances.

• In fact, the expected payoffs for two independent mixed strategies are pretty bad; if both randomly choose, the expected payoff is only 2.5 --- not much better than the maximin value.

• For your information, the set of possible payoffs for all possible combinations of mixed strategies are illustrated below.

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Shows how various combinations of mixed strategies interact.Randomly pick a mixed strategy for each, then plot the result.

Battle of the Sexes

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2,2 (4/9) 4,3 (2/9)

3,4 (2/9) 1,1 (1/9)

Shopping 2/3 Fishing 1/3

Fishing 2/3

Shopping 1/3

Wife preference

Husband preference

Suppose husband picks fishing 2/3 of the time and wife picks 2/3 shopping.

utility: 2*4/9 + 4*2/9 + 3*2/9 + 1(1/9) = 2.56

A simple competition game

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60, 60 36, 70 36, 35

70, 36 50, 50 30, 35

35, 36 35, 30 25, 25

High

High

Medium

Medium

Low

Low

Pierce

Donna

Note – no player has a dominant strategy. But low is dominated for both players. So we can predict that neither will play low. Remove it.

Iterated Elimination of Dominated Strategies• Let RiSi be the set of removed strategies for agent i• Initially Ri=Ø• Choose agent i, and strategy si such that siSi\Ri (Si

subtract Ri) and there exists si’ Si\Ri such that

• Add si to Ri, continue

• Theorem: If a unique strategy profile, s*, survives iterated elimination, then it is a Nash Eq.

• Theorem: If a profile, s*, is a Nash Eq then it must survive iterated elimination.

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ui(si’,s-i)>ui(si,s-i) for all s-i S-i\R-i

A simple competition game

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60, 60 36, 70

70, 36 50, 50

High

High

Medium

Medium

Low

Low

Pierce

Donna

Once we have removed low, medium is now a dominant strategy for both. So we predict that both Pierce and Donna will play medium.

Example – Zero Sum (most vicious)(We divide the same cake. If I lose, you win.)

Bi matrix form (show utilities separately each player)•Cake slicing•Two players

–cutter–chooser

Cutter'sUtility

Choose bigger piece

Choose smaller piece

Cut cake evenly

½ - a bit ½ + a bit

Cut unevenly

Small piece

Big piece

Chooser'sUtility

Choose bigger piece

Choose smaller piece

Cut cake evenly

½ + a bit ½ - a bit

Cut unevenly

Big piece Small piece

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Zero Sum• Scientists debate whether zero sum scenarios

really exist.

• However, many TREAT situations as if they did.

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Rationality

•Rationality–each player will take highest utility option–taking into account the other player's likely behavior

•In example–if cutter cuts unevenly

•he might like to end up in the lower right•but the other player would never do that

–-10–if the current cuts evenly,

•he will end up in the upper left–-1

•this is a stable outcome–neither player has an incentive to deviate

Cutter'sUtility

Choose bigger piece

Choose smaller piece

Cut cake evenly

(-1, +1) (+1, -1)

Cut unevenly

(-10, +10)

(+10, -10)

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Other Symmetric 2 x 2 Games• Given the 4 possible outcomes of (symmetric)

cooperate/defect games, there are 24 possible orderings on outcomes (showing preference for first player)– CC CD DC DD

Cooperation dominates– DC DD CC CD

Defect dominates Deadlock. You will always do best by defecting– DC CC DD i CD

Prisoner’s dilemma– DC CC CD DD

Chicken– CC DC DD CD

Stag hunt

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