LECTURE 2: NORMAL
FORM GAMES AND
DOMINANCE
PROF LIONEL PAGE
1
LAST WEEK
• Decisions under certainty: completeness and transitivity mean that preferences can
be represented by utility numbers.
• Decision under uncertainty: we can select dominant options and eliminate dominated
ones
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THE IMPORTANCE OF FIXING THE DOMINATING
CHOICE
• Is there any dominated choice?
 No! Neither Beach nor Park is strictly better than Mall in all states.
SUNNY RAINY
Beach 3 0
Park 0 3
Mall 2 2
A SLIGHTLY DIFFERENT BEACH-LOVING HIPSTER
• Is there a strictly dominated choice for this hipster?
 No! Beach is not as it is the best when Sunny; Mall is not as it is the best when Rainy; Park is not as
none of the other two choices is always strictly better
• But you may feel that I am just being too stringent: Although Beach is not strictly better than
Park all the time, Beach is always at least as good as Park, and sometimes strictly better.
SUNNY CLOUDY RAINY
Beach 9 8 4
Park 7 8 4
Mall 3 3 8
WEAKLY DOMINATED CHOICE
• Note 1: The dominating choice c’ has to be no worse than c in each and every state –
being better for just one state does not qualify
• Note 2: It is important to fix the dominating choice c’ – you cannot use different
dominating choices for different states
Definition
A choice, say choice c, is Weakly Dominated by
another choice c’ if and only if:
1. c’ is no worse than c in each and every state; and
2. there is at least one state in which c’ is strictly
better than c.
SO FAR…
• We have not used probability!
• Also, we have only made comparisons between choices, the magnitude of the
differences in utility numbers across choices does not matter
• In other words, the concept of dominance (dominant choice, dominated choice) does
not require:
 Probability sophistication
 Information on the magnitude of the payoffs
BUT DOMINANCE CAN ONLY TAKE US SO FAR
• Is there any (strictly or weakly) dominated choice?
 No.
 Beach is strictly the best when Sunny
 Park is strictly the best when Cloudy
 Mall is strictly the best when Rainy
SUNNY CLOUDY RAINY
Beach 10 7 0
Park 3 8 1
Mall 4 4 9
BUT DOMINANCE CAN ONLY TAKE US SO FAR
• If it is likely to be Sunny, we would expect Beach
• If it is likely to be Cloudy, we would expect Park
• If it is likely to be Rainy, we would expect Mall
• When we start using words like “likely”, we are starting to talk about probability
SUNNY CLOUDY RAINY
Beach 10 7 0
Park 3 8 1
Mall 4 4 9
PROBABILITY AND MAGNITUDE
• Even if the probability of Listeria contamination is very small, given the dire
consequence (versus the small improvement in utility when Safe), it seems
reasonable to choose Avoid
• This suggests that we cannot take utility to be ordinal anymore.
Safe Listeria Present
Eat soft cheese 2 – 1,000
Avoid soft cheese 0 0
PROBABILITY AND UTILITY
• Perhaps we can take the weighted average of the utility numbers (weighted by the
probabilities) for each choice
• Then we will pick (or predict) the choice with the highest weighted average
• This is known as Expected Utility
Sunny Cloudy Rainy
Probability 0.4 0.3 0.3 Expected Utility
Beach 10 7 0
Park 3 8 1
Mall 4 4 9
EXPECTED UTILITY: WHAT DOES IT MEAN?
• Expected utility is easy to use (this is a great advantage!)
• But what are we actually assuming when we think that a decision-maker will pick the
choice with the highest expected utility?
Sunny Cloudy Rainy
Probability 0.4 0.3 0.3
Beach Fun Breezy Wet
Park Hot Nice Soggy
Mall Meh Meh Comfy
A BIT OF TERMINOLOGY
• Given the probabilities on the states of nature, a choice leads to a probability
distribution over outcomes
• In our example, Beach leads to the following probability distribution:
• We call these distributions lotteries
Fun Breezy Wet Hot Nice Soggy Meh Comfy
Prob 0.4 0.3 0.3 0 0 0 0 0
LOTTERIES: GENERAL CASE
• In general, if there are n possible state-contingent (i.e., certain) outcomes
1, 2, … , , then a lottery can be described by:
• …where the probabilities sum up to 1.
• What economists typically do is to start with the preliminaries of how a decision-
maker compares between different lotteries
• Such comparisons are called Preferences over Lotteries
…
Prob 1 2 …
EXPECTED UTILITY PROPERTY
• Utility functions that have the expected utility property are often called von-Neumann
Morgenstern (vNM) utility functions or Bernoulli utility functions
Definition
A function V that assigns numbers to every lottery has the expected utility
property if:
= 1 1 + 2 2 + ⋯+ = �
=1

()
Where L induces a lottery defined by the probabilities p and the payoffs x.
MAGNITUDE MATTERS FOR EXPECTED UTILITY
• The expected utility property requires:
(L) = �
=1

()
• But once I started multiplying and adding, the differences between the numbers have
meanings
• The utility numbers () carry more content than mere comparisons
• In technical terms, we say that the utility . is not ordinal, but cardinal
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WHY WOULD PEOPLE USE EXPECTED UTILITY?
• Let’s denote lotteries as , , , …
• The lottery that puts probability 1 on outcome will be written simply as
• ≽ read as is weakly preferred to
ASSUMPTIONS ON PREFERENCES
• One questions we can ask:
• “What do we need to assume about the preference over lotteries, in order for us to
use expected utility?”
• These assumptions are sometimes known as axioms
 They are principles of behaviour
 If we abide by them, then we use expected utility to choose between lotteries
AXIOMS ON PREFERENCES OVER LOTTERIES:
COMPLETENESS
• Name of the Axiom Completeness
• Meaning Every pair of lotteries can be compared
• Definition For any two lotteries and ,
 Either ≽ or ≽
AXIOMS ON PREFERENCES OVER LOTTERIES:
TRANSITIVITY
• Name of the Axiom Transitivity
• Meaning Comparison of lotteries cannot go in cycles
• Definition For any lotteries , ′, ′′
 If ≽ and ≽
 Then ≽
COMPOUND LOTTERY
• So far we speak of lotteries that leads directly to certain outcomes
• But a lottery can lead to another lottery
• For example, I may say, \we’ll flip a coin,
 if it comes up head we will get pizza;
 if it comes up tail we will flip the coin again, and
o if that is head we will get pasta
o if that is tail we will fast
• A lottery that pays out only certain outcomes is called simple lottery
• A lottery that pays out another lottery is called compound lottery
AXIOMS ON PREFERENCES OVER LOTTERIES:
INDEPENDENCE
• Name of the Axiom Independence
• Meaning Common elements in two lotteries do not affect preference ordering
• Definition For all (possibly compound) lotteries , ′, and all probabilities ∈ (0,1),
 ≽ ′⇔ (, ; 1 − , ′′) ≽ (, ; 1 − , ′′)
INDEPENDENCE AXIOM: PROPERTY
• Independence axiom is not reasonable for choice under certainty
 A consumer may prefer eating cheese to eating bread
o Cheese ≽ Bread
 She may also prefers eating bread and Nutella than cheese and Nutella
o (Bread, Nutella) ≽ (Cheese, Nutella)
• Under uncertainty, perhaps it is more reasonable
o If Cheese ≽ Bread
o Then (0.5, Cheese; 0.5, Nutella) ≽ (0.5, Bread; 0.5, Nutella)
o The consumer will never have both
VON NEUMANN AND MORGENSTERN
• If a preference relation ≽ over lotteries follows completeness, transitivity,
independence (and technical axioms), then:
 It can be represented by a utility function that has the expected utility property
 = 1 1 + 2 2 + ⋯+
FOR A (DEEP) INTRO TO GAME THEORY
• With von Neumann’s role: link
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EXPERIMENT
• A: 100%, $1M
• B: 10%, $5M, 89%, $1M, 1%, $0
• A’: 11%, $1M 89%, $0
• B’: 10%, $5M, 90%, $0
EXPERIMENT
• A: 11%, $1M 89%, $1M
• B: 10%, $5M, 89%, $1M, 1%, $0
• A’: 11%, $1M 89%, $0
• B’: 10%, $5M, 89%, $0 1%, $0
ALTERNATIVES TO EXPECTED UTILITY
• The classical definition of rational choice under uncertainty is that the decision is
expected-utility-maximising. What are the modern alternatives?
 Framing
 Inability to comprehend or work with probabilities (especially when probabilities are close
to 0 or 1)
 Distrust of the experimenter
 . . . and so on
• There are alternatives to expected utility in economics (e.g., prospect theory,
ambiguity models)
• Still, Expected Utility is a good first step
FINAL REMARKS
• You might have heard of risk-aversion (or risk-neutrality, risk-loving) in another course
• And you remember that it has something to do with whether u is concave or convex
• But for game theory, we don't need those
• We will come back to this point in the next lecture, when we discuss representing
strategic interactions.
SUMMARY
1. Representing decision under uncertainty using states of nature and state-contingent
outcomes
2. Dominance
 Strictly and Weakly Dominant choices
 Strictly and Weakly Dominated choices
3. Preferences over lotteries
4. Expected Utility Theorem
 von Neumann-Morgenstern utility is no longer ordinal
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FROM DECISIONS TO INTERACTIONS
• We have looked at decision under risk
• From now on we are going to look at strategic interactions
• But what we have seen will help us a lot
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LET’S PLAY:
THE BRIEFCASE EXCHANGE
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Carmen San Diego Crackle
PAYOFFS OF THE BRIEFCASE EXCHANGE
Diamonds Nothing
Money 10 , 10 -5 , 20
Nothing 20 , -5 0 , 0
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Carmen
Crackle
Carmen’s payoffs Crackle’s payoffs
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36
THE PRISONERS'
DILEMMA
This puzzle is one of the most
famous and important games in
economics, business, law, politics,
evolutionary science, social
psychology...
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Silent Confess
Silent -1 , -1 -10 , 0
Confess 0 , -10 -5 , -5
Silent Confess
Silent -1 , -1 -10 , 0
Confess 0 , -10 -5 , -5
PAYOFF MATRIX FOR PRISONERS' DILEMMA
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Diamonds Nothing
Money 10 , 10 -5 , 20
Nothing 20 , -5 0 , 0
PRISONERS’ DILEMMA IN REAL LIFE
The Prisoners' Dilemma occurs in many fields in the real world:
1. In climate change (cutting emissions)
2. In international politics (stockpile of nuclear weapons)
3. In business (advertising budget)
4. In law (should you hire a lawyer, or not?)
5. In sport (doping)
6. In fashion and dating (make-up, cosmetic surgery)
7. In your studies (group assignments)
8. In finance (bank runs)
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PAYOFF MATRIX FOR TWO-PERSON GAMES:
GENERAL
• Payoff matrix is as easy as it can get, just remember the conventions:
1. Player 1 is always the “row player” (i.e., she chooses the row)
2. Player 2 is always the “column player” (i.e., he chooses the column)
3. The first payoff number belongs to Player 1, second to Player 2
• My own convention: Player 1 is a “she” and Player 2 a “he”
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WHAT IF WE HAVE THREE PLAYERS?
• Three students are meeting to discuss their group project
• Each can choose (without knowing others' choices) whether to prepare for the
meeting or not
• The meeting will go well only if all of them have prepared
• If at least one of them comes unprepared, the meeting will be a waste of time
• Preparing, however, takes effort and is costly (regardless of how the meeting goes)
• Each prefers a good meeting, to coming unprepared to a bad meeting (“shirking” your
responsibilities), to coming prepared to a bad meeting:
Good meeting ≻ Bad meeting, no effort ≻ Bad meeting, effort
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PAYOFF MATRICES FOR THREE-PERSON GAMES
Player 2
Prepare Shirk
Player 1
Prepare 3, 3, 3 0, 1, 0
Shirk 1, 0, 0 1, 1, 0
Prepare
Player 2
Prepare Shirk
Player 1
Prepare 0, 0, 1 0, 0, 1
Shirk 1, 0, 1 1, 1, 1
Shirk
Player 3 chooses the
payoff matrix
‘Shirk’, verb:
To avoid or neglect a duty
or responsibility.
If Player 1 chooses Shirk
…and Player 2 chooses Prepare
…and Player 3 chooses Shirk,
…then Player 1 gets 1, Player 2
gets 0, Player 3 gets 1. 42
THREE-PERSON GAMES: CONVENTIONS
1. Player 1 chooses row
Player 2 chooses column
Player 3 chooses matrix
2. The first number in any cell is the payoff to player 1
The second number in any cell is the payoff to player 2
The third number in any cell is the payoff to player 3
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MORE READINGS
The True Story of the Birth of the Prisoner's Dilemma