introducing quantitative literacy in a writing course...

Game Theory Game Practice Where’s the QL? Summary

Introducing Quantitative Literacy in a WritingCourse Using the Ultimatum Game

Rob Root

Department of Mathematics

January 13, 2015—Joint Mathematics MeetingsMAA Session on infusing QL into courses, II

San Antonio Convention Center, Room 212B

Rob Root Lafayette College

UG intro to QL


Overview

Outline1 Game Theory

BackgroundUltimatum Game

2 Game PracticeDataA New Model

3 Where’s the QL?Models MatterReasoning in real lifeExtra slides

Quantitative Literacy

The ability and habitual inclinationto use quantitative information andreasoning to understand the world,communicate with others, andmake decisions.


UG intro to QL


Background

Cultivated Ground

Books on Game Theory and QL

Rick Gillman & David Housman’s Models of Cooperation and

Competition

General education QL course using game theoryWilliam Poundstone’s Prisoner’s Dilemma (1993), Priceless (2011), &Rock Breaks Scissors (2014)

Popular books on game theory: no mathematical notation

More books, articles, and web resources


UG intro to QL


Ultimatum Game

Let’s Play a Game!

The Ultimatum Game

Rules:Stake: A prize that can be divided into tenths(in class, 10 one dollar bills)Two asymmetric players: Proposer and Responder1st, proposer proposes a split of the prize: “none for you, 10 forme;” “1 for you, 9 for me;” . . . “10 for you, none for me”Then, responder accepts proposed split or rejects itIf responder accepts, prize is won and split according to proposal.If responder rejects, prize is lost


UG intro to QL


Ultimatum Game

Classical Model

If this game is about the prize, there is only one rational outcome:Proposer offers “1 for you, 9 for me,” and so gets the most whilegiving disposer somethingDisposer accepts because one is better than noneThis is the unique Subgame Perfect Nash equilibrium (SPE)


UG intro to QL


Data

Data

What really happens

“In theory, there is no difference between theory and practice. Inpractice, there is.” — 20th century proverb

Mathematical analysis is trivialUsually at least one proposer in class performs itVery rarely do these proposers win (1 time in ⇠100 games with⇠20 attempts)Most proposers offer nearly even splits; most responders rejectsplits that greatly favor proposer


UG intro to QL


Data

Real Data

My classroom is not an experiment

“The ultimatum game is claimed to be one of the most frequentlyperformed of all human experiments today.” —William Poundstone, inPriceless

No large population of humans plays the ultimatum game usingthe SPENature of prevalent strategies varies by cultureVariations on game show this “nonrational” behavior is robust andcontextualPractice does not match theory! Now what?


UG intro to QL


A New Model

A New Model

Fairness has value

“By stripping away all the customary social, legal, financial, and ethicalentitlements, the game lays bare the issue of inequality, somethingthat all societies struggle with.” —William Poundstone, in Priceless

Instead of just the value of the prize, assign a value to equitabletreatmentNew value varies from individual to individual, could be modeledas randomRandom model can account for observed behavior post facto


UG intro to QL


Models Matter

Where’s the QL?

Models matter

“. . . the supreme goal of all theory is to make the irreducible basicelements as simple and as few as possible without having to surrenderthe adequate representation of a single datum of experience.” —AlbertEinstein

The failure of naive analysis is dishearteningBut the lesson of incomplete models is importantStudents appreciate that naive use of numbers can mislead


UG intro to QL


Reasoning in real life

You did what?!


“. . . literacies are for the most part practiced invisibly andsubconsciously . . . not pulled out selectively and applied deliberately. . . .” —Richard Ewell in Mathematics & Democracy

Responders occasionally make nonsensical choices, like rejectingeven-valued splits (10-0, 8-2, 6-4, etc)Other responders regret their choices, like accepting only anequal split and then losing.Opportunity to consider how emotion and reason interact, in a“real” situation


UG intro to QL



Whoa! Random!!

More reasoning in real life

Players confront a meaningful uncertainty via the gameMathematical models of uncertainty are rarely understood, andcounterintuitive in practiceContext is centrally importantJust as behavior in UG is sensitive to context, so is QL


UG intro to QL


Summary

UG exposes that a good model is critical for QR to be helpful, butgood models often entail much more mathematical machinerythan simpler, inaccurate onesUG demonstrates that QL is helpful understanding humanbehavior and establishing productive agreements by makingdecisions under uncertainty and in diverse contextsQR may not be the most elemental knowledge for a usefulmodel, but when understood and used in conjunction with otherknowledge, it is uniquely empowering


UG intro to QL


Extra slides

QL at the NYT?


UG intro to QL


Extra slides

Real Data

A Rich Graphic

Costly Punishment AcrossHuman SocietiesJoseph Henrich,1* Richard McElreath,2 Abigail Barr,3 Jean Ensminger,4 Clark Barrett,5

Alexander Bolyanatz,6 Juan Camilo Cardenas,7 Michael Gurven,8 Edwins Gwako,9

Natalie Henrich,1 Carolyn Lesorogol,10 Frank Marlowe,11 David Tracer,12 John Ziker13

Recent behavioral experiments aimed at understanding the evolutionary foundations of humancooperation have suggested that a willingness to engage in costly punishment, even in one-shotsituations, may be part of human psychology and a key element in understanding our sociality.However, because most experiments have been confined to students in industrialized societies,generalizations of these insights to the species have necessarily been tentative. Here, experimentalresults from 15 diverse populations show that (i) all populations demonstrate some willingnessto administer costly punishment as unequal behavior increases, (ii) the magnitude of thispunishment varies substantially across populations, and (iii) costly punishment positively covarieswith altruistic behavior across populations. These findings are consistent with models of thegene-culture coevolution of human altruism and further sharpen what any theory of humancooperation needs to explain.

For tens of thousands of years before formalcontracts, courts, and constables, humansocieties maintained important forms of

cooperation in domains such as hunting,warfare, trade, and food sharing. The scale ofcooperation in both contemporary and pasthuman societies remains a puzzle for theevolutionary and social sciences, because, first,neither kin selection nor reciprocity appearsto readily explain altruism in very largegroups of unrelated individuals and, second,canonical assumptions of self-regarding pref-erences in economics and related fieldsappear equally ill-fitted to the facts (1). Rep-utation can support altruism in large groups;however, some other mechanism is neededto explain why reciprocity should be linkedto prosociality rather than selfish or neu-tral behavior (2). Recent theoretical work

suggests that substantial cooperation canevolve, even among non-kin, in situationsdevoid of reputation or repeat interaction ifcooperators also engage in the costly punish-ment of non-cooperative norm violators (3–10).Consistent with these models, behavioralexperiments have now confirmed the (i) ex-istence of costly punishment, (ii) effective-ness of punishment in sustaining cooperation(11, 12), and (iii) willingness by uninvolvedthird parties to punish in anonymous situa-

tions (13). Such experiments have even be-gun to probe the neural underpinnings ofpunishment (14, 15).

These results are important, because theexistence of costly punishment can explainimportant pieces of the puzzle of large-scalehuman cooperation. However, like previousexperimental games used to study altruism,these experiments have been conducted al-most exclusively among university students.We do not know whether such findingsrepresent the peculiarities of students and/orpeople from industrialized societies or wheth-er they are indeed capturing species character-istics. Our earlier research used experimentalgames in 15 diverse societies to measureother-regarding behavior (1, 16). We foundthat canonical self-interest could not explainthe results in any of the 15 societies studied.We also found much more variation in gamebehavior than previous studies with universitystudents had found. Similarly, until costlypunishment is studied in more societies andoutside of university students, it is difficult tojudge its importance for explaining humancooperation.

In addition to estimating how widespreadit is, knowing whether costly punishmentcovaries with altruistic behavior is valuable.Models of the evolution of costly punishmentsuggest that societies in which costly punish-ment is common will exhibit stronger normsof fairness and prosociality, because the

1Department of Anthropology, Emory University, 1557Dickey Drive, Atlanta, GA 30322, USA. 2Department ofAnthropology, Graduate Group in Ecology, Animal BehaviorGraduate Group, Population Biology Graduate Group,University of California Davis, One Shields Avenue, Davis,CA 95616, USA. 3GPRG, Department of Economics, Univer-sity of Oxford, Manor Road, Oxford, OX1 3UQ, UK. 4Divisionof the Humanities and Social Sciences, California Institute ofTechnology, Pasadena, CA 91125, USA. 5Department ofAnthropology, UCLA, 341 Haines Hall, Box 951553, LosAngeles, CA 90095, USA. 6Department of Anthropology,College of DuPage, Glen Ellyn, IL 60137, USA. 7Facultad deEconomia, CEDE, Universidad de Los Andes, K1 No. 18A-70,Bogota, Colombia. 8Department of Anthropology, Universityof California, Santa Barbara, CA 93106, USA. 9Department ofSociology and Anthropology, Guilford College, 5800 WestFriendly Avenue, Greensboro, NC 27410, USA. 10GeorgeWarren Brown School of Social Work, Washington University,1 Brookings Drive, St. Louis, MO 63130, USA. 11Departmentof Anthropology, Harvard University, 11 Divinity Avenue,Cambridge, MA 02138, USA. 12Department of Anthropology,University of Colorado at Denver and Health Sciences Cen-ter, Post Office Box 173364, Campus Box 103, Denver, CO80217, USA. 13Department of Anthropology, Boise State Uni-versity, 1910 University Drive, Boise, ID 83725, USA.

*To whom correspondence should be addressed. E-mail:[email protected]

Fig. 1. UG results displayed as the distributions of rejections across possible offers in the UG,which overlaythe mean offers and interquartiles. For each population labeled along the vertical axis, the areas of the blackbubbles, reading horizontally, show the fraction of the sample of player 2s who were willing to reject thatoffer. For reference, inside some of the bubbles we noted the percentage illustrated by that bubble. Thedashed vertical bars mark the IMO for each population. The solid vertical bars mark the mean offer for eachpopulation,with the gray shaded rectangle highlighting the interquartile of offers. Populations were orderedby their mean offers (from low to high). Counts on the right (n) refer to numbers of pairs of players.

RESEARCH ARTICLES

www.sciencemag.org SCIENCE VOL 312 23 JUNE 2006 1767


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Extra slides

Graphic for Classroom Data

92.5% 70.% 70.% 60.% 15.% 0.% 2.5% 5.% 10.% 10.% 15.%

0 10 20 30 40 50 60 70 80 90 100

5 Years Combined


UG intro to QL

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