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Slides for Phud reunion talk

Bob Nau

Von Neumann: formalization of game theory as a foundation for economics, “utility” as a substitute for money in modeling choice under risk and strategic behavior in games

John Nash: solution concept for noncooperative games: players choose pure or independently randomized strategies to achieve an equilibrium in units of expected utility

Daniel Ellsberg: decision makers are averse to ambiguity as well as to risk and uncertainty

One of these guys is right

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At Cornell on the 30th

anniversary of publication of the Pentagon Papers, 2001 (2nd International Symposium on Imprecise Probabilities and Their Applications)

“Death and Life” by Gustav KlimtLeopold Gallery, Vienna, 2010 (at the Workshop on Risk, Ambiguity, and Decisions in Honor of Daniel Ellsberg, Institute for Advanced Studies)

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Von Neumann and Morgenstern: Theory of Games and Economic Behavior (1944/1947)

• The objective of the book is to place economics on the same footing as physics, with similar considerations of principles of measurement.

• Proposes that game theory (“the problem of 2, 3, 4, bodies…”) should be the foundation of the analysis of economic competition.

• First 45 pages, which discuss the analogies with physics and axioms of utility, are highly recommended reading. The rest goes nowhere.

• “We shall assume that the aim of all participants in the economic system, consumers as well as entrepreneurs, is money, or equivalently a single monetary commodity. This is supposed to be unrestrictedly divisible and substitutable, freely transferable, even in the quantitative sense, with whatever ‘satisfaction’ or ‘utility’ is desired by each participant.” (p. 8)

• Von Neumann and Morgenstern gave a set of axioms to be satisfied by preferences among probability distributions over an arbitrary set of personal consequences. (For example, transitivity axiom: A>B and B>C => A>C)

• The axioms imply that utilities can be assigned to consequences such that preferences are determined by maximizing expected utility.

• These utilities represent the “equivalent money” the players receive when making choices or playing games, representing their overall satisfaction.

• In general the probabilities are also subjectively determined as representations of personal beliefs, and agents update their subjective probabilities via Bayes’ Rule when they acquire information (Savage 1954).

• This combination of subjective probability and expected utility is the foundation of modern (“Bayesian”) statistics as well as game theory.

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Many models in economics rest on one of these theorems. The 2nd is the one that links rational actions of the body with rational states of mind.

Separating hyperplane theorem: two disjoint convex sets, at least one of which is open, can be separated by a nontrivial hyperplane

Fixed-point theorem: a continuous mapping of a convex compact into itself has a fixed point

• Nash (1951) proposed that the solution of a game should be an equilibriumin which each player chooses a pure or independently randomized strategy that optimizes against the other players in units of expected utility.

• Nice existence proof based on the fixed-point theorem.

• “Nash equilibrium” is the foundation of modern game theory, which in turn is now the foundation of much of economics, fulfilling von Neumann’s dream (although not via his own solution concept).

Nash equilibrium

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Nash meets von Neumann (as told by Sylvia Nasar):

Von Neumann was sitting at an enormous desk, looking more like a prosperous bank president than an academic in his expensive three-piece suit, silk tie, and jaunty pocket handkerchief. He had the preoccupied air of a busy executive. At the time, he was holding a dozen consultancies, “arguing the ear off Robert Oppenheimer” over the development of the H-bomb, and overseeing the construction and programming of two prototype computers. He gestured Nash to sit down. He knew who Nash was, of course, but seemed a bit puzzled by his visit.

He listened carefully, with his head cocked slightly to one side and his fingers tapping. Nash started to describe the proof he had in mind… But before he had gotten out more than a few disjointed sentences, von Neumann interrupted, jumped ahead to the as yet unstated conclusion of Nash’s argument, and said abruptly, “That’s trivial, you know. That’s just a fixed point theorem.”

What’s wrong with any of that?• How are the subjective parameters of preferences (personal probabilities and

utilities) observed, and if the axioms aren’t satisfied, so what?• What you can measure on a numerical scale is defined by properties of an

observer and her measuring device.• This is as true in economics as in relativity theory and quantum theory. • Different agents can’t directly observe each other’s subjective beliefs or utilities

for outcomes or their private investments. They can only observe willingness to bet or trade or to buy or sell at posted prices, enforced by contracts or markets.

• Reciprocal measurements between agents (haggling, deal-making) may also perturb their beliefs and values, resulting in fundamental indeterminacies.

• Money, the real stuff, is important. Its very role is to be the one substance that is commonly desired, divisible, transferable, substitutable, and publicly measurable in numerical units. (Why didn’t von Neumann think more carefully about this?)

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What else is wrong with Nash equilibrium?

• In general there may be more than one Nash equilibrium, and the set of them may have bizarre shapes: nonconvex, disconnected, …

• There’s no mechanism for selection or convergence.

• A 3-player game might have a unique equilibrium that involves irrational numbers. How would agents even compute them?

• They are not necessarily efficient or equitable (players might all be able to do better through some form of coordination).

My project: a unified theory of rational behavior in personal decisions, games, and markets• In all of these settings (1 body, 2-3-4… bodies, n bodies) measurements of

preferences can be performed in public in terms of buying/selling prices or side bets whose payoffs are money. If not, you don’t have observable real numbers.

• The “primal” rationality condition is that there should be no arbitrage opportunities that an outside observer successfully exploits.

• By the separating hyperplane theorem of convex analysis, there is a corresponding “dual” rationality condition in terms of maximization with respect to psychological parameters such as probabilities and utilities.

• In the most general case, probabilities and utilities are not separately measurable in absolute coordinates, as with space and time.

• Fundamental units of analysis are “risk neutral probabilities” (betting rates for money), interpretable as products of probabilities and marginal utilities.

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The correct solution concept for games (IMHO)

• Correlated equilibrium (originally due to Aumann, 1974), not Nash equilibrium.

• Arises naturally when the rules of the game are revealed through side bets.

• Randomized strategies are allowed to be correlated, although objective randomization is not required. The probabilities may be subjective.

• The set of correlated equilibria is a geometrically well-behaved object: a convex polytope (flat-sided object) whose corners have rational coordinates.

• Proof of existence is easy and constructive (fixed point theorem not needed).*

• Fundamental theorem of games (via separating hyperplane theorem): an outcome of the game does not yield an arbitrage profit to an observer if and only if it has positive probability in some correlated equilibrium. *

*Nau & McCardle 1990, Nau 1992

Example: “Battle of the sexes”Where to go for the evening? Ballet or boxing match?

Nash equilibrium #1 = TL (Alice gets her way and both go to the ballet)Nash equilibrium #2 = BR (Bob gets his way and both go to the boxing match)Nash equilibrium #3 = 2/3 T, 2/3 R (each goes to preferred venue with probability 2/3)

Why not choose to flip a coin? (a correlated equilibrium)

Payoff matrix :

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Geometry of solutions to battle-of-the-sexes:

The set of all probability distributions on 4 game outcomes is a tetrahedron. The Nash equilibria (red dots) lie on the surface of the polytope of correlated equilibria (green hexahedron) where it touches the set of independent distributions (gray saddle).* The obvious solution to this game (coin flipping) is the midpoint of the long edge.

*Nau et al. 2004

Isn’t this figure suggestive?

I’m looking for a sculptor.

Risk aversion• Typical people are averse to risk, and this is the fundamental behavioral

concept that underlies finance theory and the insurance industry.• When choosing among assets with different probability distributions of

returns, there is a “premium for risk”: you demand a higher expected return for an asset that is regarded as more risky.

• You are also willing to pay insurance premiums to reduce risk against accidental losses.

• In the standard model (Arrow-Pratt), aversion to risk is a property of your utility for money (“diminishing marginal utility”), and it is measured by a ratio of first and second derivatives of the utility function.

• This requires the agent to be equally risk averse toward all sources of risk, because the risk attitude measure depends only on utility for money.

• What’s wrong with that?

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* “Dad, what is it with all these urn problems? An urn is something you put dead people in. If you take a ball out of an urn, and there’s some ash on it, what’s the probability it’s from the femur bone? When Ellsberg dies, they should put him in an urn. They can call it Ellsberg’s urn”. --Becca Nau

Which would you prefer:$100 if ball drawn from urn A is white.$100 if ball drawn from urn B is white.

Which would you prefer:$100 if ball drawn from urn A is black.$100 if ball drawn from urn B is black.

If, like most people, you strictly prefer to draw from urn A in both cases, “you are in trouble with the Savage axioms” which require you to act as if assigning unique probabilities to all events, and to be equally averse to all risks. “Ellsberg’s urn” has spawned an enormous literature over the last 30 years.*

Ellsberg’s paradox (1961): aversion to “ambiguity” as well as risk

My resolution of the paradox:• A model of preferences in which agents have smooth indifference curves

that bend in ways that are not allowed by the standard model.

• Agents can merely be more risk averse toward some sources of risk than others (e.g., those that are more ambiguous).

• Risk aversion measure is a ratio of first and second derivatives of the risk neutral probabilities, in which utility is not separated from probability.*

• It’s directly measurable in terms of small local bets, whereas the Arrow-Pratt measure is generally unmeasurable due to unknown background risk, even when events are unambiguous.

*Nau 2001, 2006, 2011

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Statforecasting.com:> 1 million unique daily visitors per year, 20% of whom spend at least an hour

My alter ego: Mr. R-squared on the internet

Google “r-squared”, “linear regression analysis”, “time series forecasting”, “random walk model”, etc.

“User-specified-model” forecasting procedure in Statgraphics (commercial stat program), which allows side-by-side testing of up to 5 time series models built from an a-la-carte menu of data transformations and model types

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Best tool you can get for linear and logistic regression:

• Developed over the last 10 years for my teaching, released to the public in 2014• 40-button ribbon interface for model space navigation and output exploration• Interactive, presentation-ready table and graph output• Built-in teaching notes and other features to support teaching (e.g., mass grading)• 2-way interface with the R programming language: allows Excel to be used as a front end

and/or back end for running larger models in R with customized output in RStudio and Excel.

Available for free at regressit.com

Logistic regression example: whois predicted to die on the Titanic?