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NORMAL FORM GAMES AND DOMINANCE

创建日期：2024-06-12 15:10:43

所属分类：计算机computer science

标签： code

NORMAL FORM GAMES AND DOMINANCE

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LECTURE 2: NORMAL

FORM GAMES AND
DOMINANCE
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
2
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
Go to www.menti.com and enter the code:
3268 8499
1 question
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

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