government 2005: formal political theory i · 4.voters observe outcome ( or zero) and decide...

Government 2005: Formal Political Theory ILecture 12

Instructor: Tommaso NanniciniTeaching Fellow: Jeremy Bowles

Harvard University

November 16, 2014

Overview

I Political agency and accountability (Besley 2006)

I Cheap talk games (Crawford and Sobel 1982)I Politicians and policy advisers

Political accountability

* Setup

I Two periods: t ∈ {1, 2}I Politician set policy: et ∈ {0, 1}I State of the world: st ∈ {0, 1}I Voters’ payoff: ∆ > 0 if et = st , zero otherwise

I Discount factor (common to politician and voters): β < 1

I Two types of politician, congruent vs dissonant: i ∈ {c , d},with prior equal to Prob[i = c] = π

I Both types get E (ego rents) when in office

I c-type also gets ∆ when et = stI d-type also gets rt when et = 1− st

I rt ∼ G (r) in [0,R] with E (r) = µI We assume R > β(µ+ E ) (we’ll see why)

I Hence, politician’s strategy: et(s, i)

Political accountability (contd.)

* Timing

1. Nature decides i of incumbent politician and s (bothunobserved to voters)

2. Nature draws r1 from G (r)

3. Incumbent decides e1

4. Voters observe outcome (∆ or zero) and decide whether toreelect the incumbent or to draw a new politician (congruentwith probability π)

5. Nature draws r2 from G (r) and the politician in office decidese2; payoffs are determined


I What are the PBE of the model?I In period 2, we simply have that:

I e2(s, c) = s2I e2(s, d) = 1− s2

I In period 1, we have that:I e1(s, c) = s1 (congruent politician always do what voters want

provided they reelect him for doing so)I Prob[e1(s, d) = s1] = λ (index of political discipline of

dissonant politician)

I Voters’ posteriors:

π̂ = Prob[i = c |e1 = s1] =π

π + (1− π)λ≥ π

ˆ̂π = Prob[i = c |e1 = 1− s1] = 0 < π

I Hence, if ∆ observed, incumbent is reelected (sequentiallyrational behavior by voters)


I What’s the best response by a dissonant politician? (λendogenous)

I e1 = s1 iff r1 < β(µ+ E )

I As a result, λ = G (β(µ+ E ))

I As we assumed R > β(µ+ E ), ICCd is met to sustainseparating outcome for at least some dissonant politicians

I What happens if R ≤ β(µ+ E )?I We have identified PBE:

I e∗k (s, c) = skI e∗2 (s, d) = 1− s2I e∗1 (s, d) = s1 if r1 < β(µ+ E )I e∗1 (s, d) = 1− s1 if r1 ≥ β(µ+ E )I ˆ̂π = 0I π̂ = Prob[i = c |e1 = s1] = π

π+(1−π)G(β(µ+E))


I Are there other PBE?

I No complete pooling on e1 = s1 is possible (high-rentdissonant politician cannot be convinced by any belief)

I But what about pooling on e1 = 1− s1?

I To sustain this equilibrium, we must have:Prob[i = d |e1 = s1] = 1 off the equilibrium path

I So that the congruent politician is not reelected when playinge1 = s1

I In this case, e∗1(s, c) = 1− s1 iff: (1− β(1− π))∆ < βE

I This is another PBE

I It is easy to see, however, that this pooling PBE doesn’tsurvive the intuitive criterion


* Welfare analysis

I V1(λ) = [π + (1− π)λ]∆

I V2(λ) = π[1 + (1− π)(1− λ)]∆

I W (λ) = V1(λ) + βV2(λ)

I W increasing in λ

I W increasing in π

I Negative correlation between welfare and political turnover(i.e., (1− π)(1− λ))

I Comparing V1 and V2, positive analysis of term limit:discipline effect vs selection effect

Cheap talk games

I The structure of cheap talk games is the same as that ofsignaling games:

1. Nature draws a type t for the sender2. The sender observes t and chooses a message m3. The receiver observes m (but not t) and chooses an action a4. Payoffs US(t, a) and UR(t, a) are determined

I The difference is in the payoffs → the messages sent by thesender do not directly affect the payoffs of either thesender or the receiver

I Payoffs depend only on the sender’s type and the receiver’saction

I Cheap talk is really cheap, that is, costless, non-binding, andnon-verifiable

Cheap talk games (contd.)

I The payoffs of the sender and receiver must satisfy threenecessary conditions in order for cheap talk to be informative:

1. Different sender types have different preferences over thereceiver’s actions

2. The receiver prefers different actions depending on the sendertype (same condition in signaling games)

3. The receiver’s preferences over actions are not completelyopposed to the sender’s preferences

I The sender and receiver must have some common interests

I We can characterize pooling (uninformative) equilibria, aswell as separating or partially separating equilibria (wheresome information about the sender’s type is conveyed)

Cheap talk games (contd.)

I In cheap talk games there is always a pooling (uninformative)equilibrium, in which the messages are ignored by the receiverand all senders send the same message

I This is sometimes called a “babbling” equilibrium

I The receiver believes that all sender types will send the samemessage

I Off-equilibrium beliefs must ensure that all sender types sendthe same message

I E.g., the receiver believes that, if a sender deviates, she musthave an average type

I Then, the message is uninformative and the receiver’sposterior belief is equal to his prior belief

I We have an equilibrium since nobody has an incentive todeviate

Cheap talk games (contd.)Example 1

I Let’s consider the following payoff examples, with two sendertypes tL and tH , and two actions aL and aH

tL tHaL (2,1) (1,0)

aH (0,0) (0,1)

I Note: This is a payoff matrix but not as a function of player’sactions

I The receiver want to play aL with tL and aH with tHI However, type-L and type-H both prefer the action aL to the

action aHI So, both senders want to send the message t = tLI The receiver cannot believe the tL sender’s message

I The first condition discussed above is violated


tL tHaL (0,1) (1,0)

aH (2,0) (0,1)

I The preferences of the sender and receiver are diametricallyopposed

I When the sender’s type is L, the sender prefer the action aHbut the receiver prefers aL

I When the sender’s type is H, the sender prefer the action aLbut the receiver prefers aH

I The sender always wants the receiver to be deceived about histype

I So, the receiver cannot believe the sender’s message

I The third condition discussed above is violated


tL tHaL (2,1) (0,0)

aH (0,0) (1,1)

I The preferences of the sender and receiver are perfectlyaligned, and the sender can truthfully reveal his type

I Denote q and p as the belief that a high type sent themessage m(tL) and m(tH), respectively

I The following is a perfect Bayesian equilibrium:[(m(tL),m(tH)), (a(m(tL)), a(m(tH))), (q(m(tL)), p(m(tH)))] =[(tL, tH), (aL, aR), (0, 1)]

I All conditions discussed above are met

Politicians and policy advisers

I Let’s consider a more general cheap talk model with acontinuum of types/messages

I A politician/decision-maker (the receiver) must choose apolicy (the action) from the interval [0, 1]

I An expert/adviser (the sender) has information about whatthe best policy is

I The adviser does not have exactly the same preferences of thepolitician—rather, the adviser always prefers policies that areslightly higher

Politicians and policy advisers (contd.)

I More precisely, the sender’s type is t and t ∼ U[0, 1]

I An alternative way to see this game is that t is the state ofthe world, which the expert learns about

I Denote the policy chosen by the politician as a

I The politician’s payoff is −(a− t)2

I The adviser’s payoff is −(a− t − b)2, where b ≥ 0

I Then, t is the bliss point of the politician, and t + b the blisspoint of the adviser

I If the sender’s type is t, then the sender has received privateinformation about the best policy for each player


I When the sender’s type is t, the politician has quadraticpreferences with ideal point at t, and the adviser has quadraticpreferences with ideal point at t + b, as depicted below

t+b

US(t,a)

t

UR(t,a)

I The larger is b, the greater is the adviser’s “bias” in favor ofhigher policies

I When b is close to 0, then the interests of the politician andthe adviser are closely aligned


I All perfect Bayesian equilibria are equivalent to partiallypooling equilibria of the following form

I There are n ≥ 1 intervals [0, x1), [x1, x2), ..., [xn−1, 1] such thatall types in the same interval convey the same message, buttypes in different intervals convey different messages

I The pooling/babbling equilibrium, with just one interval, isthe case with n = 1

I We might assume that the message sent by the types in theinterval [xk , xk+1) is simply “t is in [xk , xk+1)”

I There is a maximum number of intervals, which depends on b

I Crawford and Sobel (1982) show that when b → 0, n→∞,and thus, there is perfect separation


I To see how this works, consider the PBE with a 2-intervalequilibrium, i.e., n = 2

I We must find a point x1 such that:I All types with t ∈ [0, x1) prefer sending the message “t is in

[0, x1)” to the message “t is in [x1, 1]”I All types with t ∈ (x1, 1] prefer sending the message “t is in

[x1, 1]” to the message “t is in [0, x1)”I The receiver updates his belief about the sender’s type using

Bayes’ ruleI And chooses policy to maximize his expected payoff, given his

updated belief


I Given the message “t is in [0, x1)” Bayes’ rule implies that thereceiver’s posterior belief is that t ∼ U[0, x1)

I That is, f (t) = 1/x1 for t < x1, and f (t) = 0 for t ≥ x1. Thereceiver’s optimal action is then a = x1/2

0 1x1/2 (x1+1)/2x1

I Similarly, given the message “t is in [x1, 1]” the receiver’sposterior belief is that t ∼ U[x1, 1], and his optimal action isthen a = (x1 + 1)/2


I If we are looking at an equilibrium, then all senders witht ∈ [0, x1) must prefer the policy x1/2 to the policy (x1 + 1)/2

I Also, all senders with t ∈ [x1, 1] must prefer the policy(x1 + 1)/2 to the policy x1/2

I Since the sender’s preferences are symmetric about his idealpoint, he prefers x1/2 to (x1 + 1)/2 iff x1/2 is closer to hisideal point than (x1 + 1)/2

I This is true iff t + b is less than the midpoint between x1/2and (x1 + 1)/2

midpoint0 1x /2 (x +1)/2midpoint0 1x1/2 (x1+1)/2

t+b

US(t,a)US(t,a)


I The sender’s preferences are continuous in his type

I So, at an equilibrium, a sender with type t = x1 must beindifferent between the policies x1/2 and (x1+1)/2

I That is, (x12− x1 − b

)2=

(x1 + 1

2− x1 − b

)2

−(x1

2− x1 − b

)=

x1 + 1

2− x1 − b

2(x1 + b) =x1 + 1

2+

x12

x1 =1

2− 2b


I First, note that x1 is uniquely determined for a given bias b

I Second, note that x1 ≥ 0 iff b ≤ 14

I That is, if b is too large, then only babbling can occur inequilibrium

I Useful communication is only possible when b is smallenough, that is, when the preferences of the politician and hisadvisor are not too dissimilar

I Third, x1 <12 , so the first interval is shorter than the second

I Thus, in a sense, and on average, advisers whose preferencesare closer to those of the politician send “more informative”messages than advisers whose preferences are farther away


* Possible remedies:

I Extensive communicationI Delegation

I If policy bias not too large, delegation is better than any cheaptalk equilibrium

I ContractsI Contracts are very effective but costly for the politician (i.e.,

full revelation is always feasible but never optimal)

I Multiple sendersI How should the politician extract information, with

simultaneous or sequential talks? Divide and rule?

Government 2005: Formal Political Theory ILecture 13

Instructor: Tommaso NanniciniTeaching Fellow: Jeremy Bowles

Harvard University

November 30, 2017

Tales from the Lab

I Experiments in social sciences

I Behavioral game theory

I Two applications:

1. Property rights in ultimatum and dictator games2. Evolution of cooperation in repeated games

I Lab vs field experiments

Experiments in social sciencesDo they make sense?

The author is grateful for the support of the Economic and Social Research Council.i

This work was undertaken as part of the Economic Beliefs and Behaviour Programme, awardnumber L 122 251 024. Many of the ideas discussed in this paper have developed as a result ofdiscussions with colleagues at the University of East Anglia including Robin Cubitt, Martin Hollis,Judith Mehta and Robert Sugden.

Experiments in Economics ...

(should we trust the dismal scientists in white coats?)

By

Chris Starmer , i

School of Economic and Social Studies,UEA, Norwich, UK.

tel: 01603 592899

e-mail: C. [email protected]

"unfortunately, we can seldom test particular predictions in the social sciences byexperiments explicitly designed to eliminate what are judged to be the most importantdisturbing influences." M. Friedman (1953) "The Methodology of Positive Economics".

"It is rarely, if ever, possible to conduct controlled experiments with the economy. Thuseconomics must be a non-laboratory science." R. Lipsey (1979), An Introduction toPositive Economics Fifth Edition.

".... social scientists rarely, if ever, are in a position to repeat any experiment; we typicallytake the data as given and, in many applications, are not even in a position to isolatephenomenon under consideration from the general economic environment." J. Darnell andJ. Lynne Evans (1990), The Limits of Econometrics.

"One of the weaknesses in the claim that the social sciences are sciences at all is theirinability to conduct controlled experiments. Physicists can create vacuums, chemists canestablish sterile environments, even doctors can conduct blind trials. But economists,sociologists political scientists and those who study management find their subject matterwill never stand still.” John Kay,, Financial Times, 3rd Jan,1997.

Experiments in social sciences (contd.)A little bit of (selected) history

* Source: Roth, A. (1995), “An Introduction to ExperimentalEconomics,” The Handbook of Experimental Economics Vol.1

I Thurstone (1931) experiment on indifference curves

I The Wallis-Friedman (1942) critique

“It is questionable whether a subject in so artificial an experimental

situation could know what choices he would make in an economic

situation; not knowing, it is almost inevitable that he would, in entire

good faith, systematize his answers in such a way as to produce plausible

but spurious results.”

“For a satisfactory experiment it is essential that the subject give actual

reactions to actual stimuli. Questionnaires based on conjectural

responses to hypothetical stimuli do not satisfy this requirement.”

Experiments in social sciences (contd.)A little bit of (selected) history (contd.)

I Dresher-Flood (1950s) experiments on prisoner’s dilemma

I Nash (1958) comment

“The flaw in this experiment as a test of equilibrium point theory is that

the experiment really amounts to having the players play one large

multimove game. There is much too much interaction, which is obvious

in the results of the experiment. Viewing it as a multimove game a

strategy is a complete program of action.”

“It is really striking, however, how inefficient Row and Column were in

obtaining the rewards. One would have thought them more rational.”

“If this experiment were conducted with various different players rotating

the competition and with no information given to a player of what

choices the others have been making until the end of all the trials, then

the experimental results would have been quite different.”

Experiments in social sciences (contd.)A little bit of (selected) history (contd.)

I Schelling (1957) on focal points

Experiment 1: You and your partner (rival) are to be given $100 if you

can agree on how do divide it without communicating. Each of you is to

write the amount of his claim on a sheet of paper; and if the two claims

add to no more than $100, each gets exactly what he claimed. If the two

claims exceed $100, neither of you gets anything

Experiment 2: You and your two partners (or rivals) each have one of the

letters A, B, and C. Each of you is to write these three letters, A, B, C,

in any order. If the order is the same, you get prizes totaling $6, of which

$3 goes to the one whose letter is first on all three lists, $2 to the one

whose letter is second, and $1 to the person whose letter is third. If the

letters are not in identical order, none of you gets anything

I In the first, 36 out of 40 subjects chose $50

I In the second, 9 out of 12 A’s, 10 out of 12 B’s, and 14 out of16 C’s chose the order ABC

Experiments in social sciences (contd.)What did we learn from early experiments?

I Bring the money

- Experiments in which subjects’ behavior determines how muchthey earn (unlike social psychology)

I Strict (and bilateral) interplay with theory

- From theory to experiments, and vice versa

I The devil is in the detail

- Non-repeated, non-symmetric, anonymous interactions- Relevance of framing, context, information on knowledge of

the game

Experiments in social sciences (contd.)Some examples/series

I Free-riding games

I Coordination games

I Bargaining gamesI Experiments in political economy

* Reference: Palfrey, T. (2016), “Experiments in PoliticalEconomy,” The Handbook of Experimental Economics Vol.2

Experiments in social sciences (contd.)Pros

I Subjects are randomly assigned to the treatment conditions(no selection bias)

I Ceteris paribus analysis of motivated individual agents (noendogeneity bias)

I Variables that cannot be directly observed in the field can beobserved in the lab (e.g., reservation wages, anticipated versusnon-anticipated money supply shocks)

I Better direct control as a substitute for complicatedeconometric methods

I Statistical tests are replicable under similar conditions

Experiments in social sciences (contd.)Cons

I External validity (Campbell and Stanley, 1963): To whatpopulations, settings, treatment variables, and measurementvariables can this effect be generalized?

I Induction problem: Behavioral regularities persist in newsituations only if the relevant underlying conditions areunchanged

I Representativity problem: Experimental subjects may or maynot be representative of out-of-sample populations

I Related accusations:I Participants are just studentsI The stakes are smallI The number of participants is smallI Participants are inexperienced

Behavioral game theoryApproach

* References: Camerer, C. (1997), “Progress in BehavioralGame Theory,” Journal of Economic PerspectivesRabin, M. (1998), “Psychology and Economics,” Journal ofEconomic Literature

I Describe actual behavior

I Driven by empirical observation

I Middle course between over-rational equilibrium analysis andunder-rational adaptive analysis

I Draw insights from psychological research

Behavioral game theory (contd.)(Some) important results

I Other-regarding preferences

I Fairness equilibrium

I Loss aversionI Cognitive biases

I Confirmation biasI OverconfidenceI Small sample bias

I Framing

I Knowledge of the game (e.g., first mover)

Property rights in ultimatum and dictator games

I Usually, in ultimatum and dictator games, participants seemto ask the question: Is this allocation fair?

I In ultimatum, more than 50% of the times, offers below 20%are rejected

I Anticipating this, usual range of offers between 40% and 50%

I In dictator, less generous but non-zero offers from 20% to 30%

I BUT perceptions on property rights matter too!(See Hoffman, McCabe, and Smith, Ch.47 Handbook Vol.1)

I 12 subjects per sessionI Random/divide treatment: Pair is “provisionally allocated”

$10 to be split with one-shot ultimatum or dictator gameI Contest/divide treatment: 6 first movers are chosen with

general knowledge quizI (Additional treatment: “seller-buyer” exchange-like framing)

Property rights in ultimatum and dictator games (contd.)Ch. 47: Preferences and Property Rights in Ultimatum and Dictator Games 419

(a)

(b)

(c)

Figure 2. (a) Replication in which 24 pairs use Forsythe et al. (1994) instructions for the $10 ultimatumgame with random assignment of the right to propose a division of the $10. (b) Treatment with 24 pairs usingForsythe et al. (1994) instructions, but in each of four sessions with 12 subjects the right to propose a divisionof the $10 is earned by being among the top 6 scores on a general knowledge quiz. (c) 24 pairs use instructionswhich formulate the $10 ultimatum game as an exchange between a buyer and a seller (see Figure 1). In eachpair the right to be the seller-proposer is assigned at random. (d) 24 pairs use the buyer/seller exchangeinstructions, but the right to be the seller-proposer is earned by scoring highest on the general knowledge test

as above.

3. Ultimatum Results

Figure 2 presents the results for all four experimental treatments. The green bars repre-sent the percentage of accepted offers; the red bars represent the percentage of rejectedoffers. Where statistical significance is reported, a Wilcoxon rank-sum test is used tocompare the sample distributions of all offers across treatments.

Property rights in ultimatum and dictator games (contd.)Ch. 47: Preferences and Property Rights in Ultimatum and Dictator Games 421

(a)

(b)

Figure 3. (a) Replication in which 28 pairs use Forsythe et al. (1994) instructions for the $10 dictator gamewith random assignment of the right to dictate a division of the $10. (b) Treatment in which 24 pairs useinstructions that formulate the $10 dictator game as an exchange between a buyer and a seller. In each pair

the right to be the seller-dictator is earned by scoring highest on a general knowledge quiz.

and there are no offers of $5. Consequently, the shift in shared expectations, and be-havior, with the implied property right treatments in ultimatum games carries over todictator games.

5. Discussion

These results demonstrate quite clearly the impact of property rights on the shared ex-pectations about appropriate proposer behavior in both ultimatum and dictator games.If the game is presented as division (which by one definition means literally to separateinto equal parts by a divisor) and players are randomly assigned to positions, there isno legitimate reason why the proposer should be allowed to exploit his or her strategicadvantage. The norm of equality applies and both players assume that is the operatingnorm. Thus, in the ultimatum game, the proposer knows that deviations from equal divi-sion may be punished. In the dictator game, less than 20% of the dictators feel justifiedin leaving $0 and 25% still feel compelled to give $5.

However, when the game is presented as a market, with the proposer named the seller,the norm of equity allows the seller to earn a “profit” on the exchange. Similarly, whenthe proposer must “earn the right” to be a proposer, the norm of equity allows the pro-

Evolution of cooperation in repeated games

I Problem with infinitely repeated games: multiplicity ofequilibria

I Dal Bo and Frechette (“The Evolution of Cooperation inInfinitely Repeated Games: Experimental Evidence,” AmericanEconomic Review, 2011) address the issue in the lab

I They (exogenously) manipulate continuation probability andpayoffs

I Each subject participates in between 23 and 77 infinitelyrepeated games

I Aim: To study how cooperation evolves as subjects gainexperience

I Main result: Being a possible equilibrium action is anecessary but not sufficient condition for cooperation toarise with experience

Evolution of cooperation in repeated games (contd.)

5

infinitely repeated games played by finite automata and find that the set of possible

payoffs depends crucially on the definition of ESS and the way costs of complexity are

modeled (also see Fudenberg and Maskin, 1990 and 1993). In contrast, Volij (2002)

shows that always defecting is the unique stochastically stable strategy (Michihiro

Kandori et al., 1993, and Peyton H. Young, 1993) in games with finite automata. Philip

Johnson, David K. Levine and Wolfgang Pesendorfer (2001) study stochastically stable

strategies in a random matching gift-giving repeated game with local information

systems. They find that cooperation (gift-giving) is stochastically stable only if the payoff

from cooperation is above a critical value that exceeds what is required by sub-game

perfection (see also Levine and Pesendorfer, 2007). Finally, Blonski and Spagnolo (2001)

appeal to the concept of risk dominance as an equilibrium selection criteria in infinitely

repeated games.

This variety of theoretical results underscores the need for empirical data to solve

the issue of multiplicity of equilibria in infinitely repeated games. The experimental

results we present can inform theories. Theories in which subjects always coordinate on

defection, even when they are infinitely patient, and theories in which they will always

coordinate on cooperation are not supported by the data. However, we find empirical

support for theories predicting cooperation under sufficiently favorable conditions.

I. Experimental design

We induce infinitely repeated game in the lab by having a random continuation

rule: after each round the computer decided whether to finish the repeated game or have

an additional round depending on a random number. We consider two probabilities of

continuation: δ=1/2 and δ=3/4. The stage game is the simple prisoners’ dilemma game in

Table 1 where the payoffs are denoted in points (one point equals to $0.006) and where

the payoff to cooperation takes one of three possible values: R=32, 40 and 48.

Table 1: Stage Game Payoffs C D

C R, R 12, 50

D 50, 12 25, 25

6

Therefore we have two main treatment variables, the probability of continuation

and the payoff from cooperation, resulting in a total of six treatments. In each session, a

set of subjects participated anonymously through computers in a sequence of infinitely

repeated prisoners’ dilemma games. Subjects were randomly re-matched with another

subject after the end of each repeated game.4 In each session subjects participate in as

many repeated games as was possible such that the first repeated game to end after 50

minutes of play marks the end of the session. The probability of continuation and the

payoff matrix was the same for all repeated games in a session, that is, there was one

treatment per session. We conducted three sessions per treatment. The instructions for

one of the sessions are in the appendix.

The treatments and results are organized around three questions that derive from

the theoretical background described next.

II. Theoretical Background

If we assume that the payoffs in Table 1 are the actual total payoffs the subjects

obtain from the stage game and that this is common knowledge, the set of subgame

perfect equilibria can be calculated as in Stahl (1991). Table 2 indicates those treatments

under which cooperation can be supported as a sub-game perfect equilibrium action. 5

Table 2: Cooperation as Equilibrium (SGPE) and Risk Dominant (RD) Action R=32 R=40 R=48

δ =1/2 Neither SGPE or RD SGPE SGPE and RD

δ=3/4 SGPE SGPE and RD SGPE and RD

4 Random matching allows for a larger number of repeated games in a session than alternative matching protocols like turnpike protocols. While the probability of a pair of subjects interacting together in more than one repeated game is high this is not likely to be a problem for several reasons. First, our results in section III suggest that the matching protocol does not introduce additional repeated games effects –for example, cooperation reaches one-shot levels when it cannot be supported in equilibrium. Second, Dal Bó (2005) uses a turnpike protocol with results consistent with other studies that have used random matching protocols. Third, with sessions with similar number of subjects Duffy and Ochs (2009) find that random matching is not enough to develop cooperative strategies across matches. 5 More precisely, the critical value of δ over which cooperation can be supported in equilibrium is 0.72 under R=32, 0.4 under R=40, and 0.08 under R=48. While this categorization of treatments is done assuming risk neutrality of the subjects, the results of the paper are robust to considering risk aversion levels of the magnitude typically observed in experiments.


I They address three questions:

1. Do subjects learn to defect when it is the only equilibriumaction?

2. Do subjects learn to cooperate when it is one of the possibleequilibrium action?

3. Do subjects learn to cooperate when it is risk dominant?


11

Table 4: Percentage of Cooperation by Equilibrium Condition and Risk Dominance

Repeated Game First Rounds All Rounds Begins Cooperation is Cooperation is

in Interaction Not SGPE SGPE Not SGPE SGPE All Not RD RD All Not RD RD

1-10 28.57 39.11 31.43 46.53 21.00 34.42 23.56 42.11 11-20 13.04 28.54 20.60 36.26 12.91 27.19 18.10 35.09 21-30 12.23 31.01 14.86 44.34 11.97 33.61 13.48 45.36 31-40 10.61 36.04 14.01 51.83 10.51 38.64 14.63 52.72 41-50 10.20 34.88 14.21 53.99 7.85 34.98 13.81 53.09 51-60 9.75 41.47 18.51 57.47 6.54 39.85 16.32 61.30 61-70 7.14 37.89 17.54 48.98 8.09 40.02 19.21 54.44 71-80 5.65 36.86 20.32 50.00 4.48 39.73 19.10 55.99 81-90 4.72 38.60 20.57 58.42 6.20 44.39 20.75 60.89 91-100 6.11 40.91 22.01 54.88 7.91 47.11 19.28 66.45

101-110 6.64 45.38 17.93 67.62 11.99 46.12 19.50 66.92 111-120 5.50 49.77 22.46 70.61 6.45 55.88 22.60 73.86 121-130 5.77 45.95 21.03 62.05 11.11 43.31 21.99 59.60 131-140 8.33 47.43 30.70 59.49 9.17 42.99 26.23 61.40

141- 46.32 23.86 65.69 47.83 16.57 76.82

From the aggregated data in this treatment it is clear that subjects learn to defect

and cooperation reaches negligible levels when cooperation cannot be supported in

equilibrium. We reach a similar conclusion when we study the evolution of cooperation

in each session under this treatment. Figure 1 displays the proportion of cooperation in

the first round of each repeated game by session and treatment. The first graph in Figure

1 displays the evolution of cooperation for the three sessions with δ=1/2 & R=32. It is

clear from this graph that cooperation decreases with experience in all three sessions.

C. Do subjects learn to cooperate when it is an equilibrium action?

The second column in Table 4 shows the percentage of subjects that choose to

cooperate in the first round of the repeated games under which cooperation can be

supported in sub-game perfect equilibrium. Initially, cooperation was 39%, but in

repeated games between 111 to 120 interactions cooperation increased to 50% (p-value of

the difference is 0.11). In the sixth column in Table 4 we observe a similar evolution of

cooperation from all the rounds in the repeated games (p-value = 0.004). In addition,

cooperation rates differ significantly depending on whether cooperation can be supported

in equilibrium for the first 10 interactions (p-value = 0.083 for round 1 only and 0.028 for


17

detailed description of the estimation procedure is in the online appendix. The estimates

of the proportions for each strategy are presented in Table 7 (with the coefficient for T2

being implied by the fact that the proportions must sum to one and gamma captures the

amount of noise – as gamma goes to infinity response become purely random).

Table 7 reveals some interesting patterns. First, as expected, cooperative

strategies describe the data better in treatments where more cooperative behavior is

observed. Second, the cooperative strategy that is most often identified is TFT. While G

explains some of the data, its proportion is not statistically significant. Finally, only

considering AD and TFT can account for 80% of all the data in matches that start after

interaction 110. Moreover, for all treatments it cannot be rejected at the 5% level that

subjects only use AD or TFT. At the 10% level, this can only be rejected under two

treatments (δ=1/2 & R=48 and δ=3/4 & R=40). Therefore, in what follows we focus on

these two strategies.

Table 7: Estimation of Strategies Used δ=1/2 δ=3/4

R = 32 R = 40 R = 48 R = 32 R = 40 R = 48 AD 0.920*** 0.783*** 0.533*** 0.648*** 0.109 0.000 (0.085) (0.074) (0.109) (0.119) (0.096) (0.000) AC 0.000 0.078 0.072 0.000 0.296** 0.079 (0.000) (0.059) (0.046) (0.000) (0.123) (0.085) G 0.000 0.040 0.000 0.000 0.267 0.116 (0.000) (0.040) (0.000) (0.024) (0.202) (0.195) TFT 0.080 0.098 0.376*** 0.352*** 0.327* 0.561*** (0.085) (0.070) (0.112) (0.115) (0.186) (0.185) WSLS 0.000 0.000 0.019 0.000 0.000 0.000 (0.000) (0.007) (0.026) (0.000) (0.000) (0.000) T2 0.000 0.000 0.000 0.000 0.000 0.244 Gamma 0.362*** 0.541 0.428*** 0.447*** 0.435*** 0.287*** (0.098) (1.077) (0.061) (0.053) (0.126) (0.061) Bootstrapped standard errors in parenthesis.

We now study how well a learning model does in fitting the evolution of

cooperation in our experiments. Then, we use the estimates of this model to perform

simulations showing that the results of this paper would also hold in the very long run.

Lab vs field experiments

* Reference: Levitt, S.D. and List, J.A. (2007), “What DoLaboratory Experiments Measuring Social Preferences RevealAbout the Real World?” Journal of Economic Perspectives

I Note the original title of the paper: “What Do LaboratoryExperiments Tell Us About the Real World?”

I Thesis: Human behavior may be sensitive to a variety offactors that systematically vary between the lab and the realworld (this may be true also in field experiment, though)

I Model: Besides monetary calculations, human decisions areinfluenced by

I Whether actions are scrutinized by othersI Context in which a decision is embeddedI Self-selection of individuals making the decision

I Final take home: Lab and field experiments as complementsmore than substitutes

government 2005: formal political theory i · 4.voters observe outcome ( or zero) and decide...

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