Coordinating Contributions to a Threshold Public Good

in a Noisy Best Response Framework

Daniel K. Saunders∗

University of California, Santa Barbara

December, 2012

Abstract

In contrast to standard public goods that suffer from the free-rider problem, thresh-

old public goods have many efficient equilibria; resulting in a coordination problem. Al-

though refinements that focus on these efficient equilibria predict successful provision,

regardless of group size, several experiments show that coordination is more difficult

in large groups. Theory also implies that incomplete information will harm provision,

while experiments find little evidence of this. I reconcile this disparity between predicted

and observed behavior using the logit quantal response equilibrium. I conduct an ex-

periment to estimate the noisiness of best response according to the logit parameter,

and interpret its effect on the rate of provision across group size and information. The

comparative statics for the provision rate predicted by the LQRE model largely bear

out in the experiment, except that incomplete information actually increases provision

among large groups. Hence, if players are boundedly rational and face a sufficiently

complex coordination problem, complete information can actually harm welfare.

J.E.L. Codes: C92, D03, H41

Keywords: provision point, public goods, quantal response, bounded rationality

∗Email: saunders@econ.ucsb.eduFunding for this research was provided by the department of economics at UCSB. The experiment washosted by the Experimental and Behavioral Economics Laboratory at UCSB. Special thanks to TheodoreL. Turocy of The Gambit Project (McKelvey et al., 2010) for his generosity with programming advice andPython code. The experiment was conducted with the software z-Tree (Fischbacher, 2007).

1

1 Introduction

Many privately provided public goods make use of a threshold, or provision point, to deter-

mine whether the good is produced. Notable examples include unitary public goods, such

as a bridge or a park, or non-unitary goods that make use of a provision point mechanism,

such as the crowd funding websites www.kickstarter.com and www.indiegogo.com that use

thresholds to determine whether a good is produced. Under complete information, the un-

dominated, trembling hand perfect Nash equilibrium refinement guarantees provision at the

threshold (Bagnoli and Lipman, 1989), and all equilibria that survive the refinement are ef-

ficient, regardless of group size. However, several experiments observe that large groups are

slower to converge to equilibrium play in laboratory experiments.1 This behavioral anomaly

is consistent with the intuition that players struggle to solve the underlying coordination

problem. Additionally, I show how standard theory predicts that incomplete information

will decrease the provision rate, while Marks and Croson (1999) are unable to find such

an effect in their experiment. This divergence between prediction and observation, across

group size and information, constitutes a puzzle.

I resolve this contradiction using the logit quantal response equilibrium model of noisy

best response. The coordination issue arising from multiple equilibria may be exacerbated

if players are boundedly rational. If players are sufficiently noisy decision-makers, than

strategic uncertainty may swamp informational uncertainty, so it is important to disentangle

these two sources of payoff uncertainty. I achieve this using the logit parameter to measure

decision noise across group size and information treatments. Rather than motivate this

model through misperceived utility, as in structural QRE (McKelvey and Palfrey, 1995), I

employ the model of noisy directional learning developed by Anderson et al. (2004). In this

framework, players dynamically adjust their behavior in the direction of higher expected

payoffs, subject to a normal error. This approach has a natural intuition, since players’

1Bagnoli and McKee (1991) find slower convergence to equilibrium play for large groups. In addition,Goeree et al. (2005a) study the Volunteer’s Dilemma, a parameterization of the provision point in whichat least one player’s valuation of the public good exceeds the threshold. They derive and reject a mixed-strategy Nash equilibrium explanation of group size effects under complete information, and suggest noisybest response and inequity aversion as an alternative explanation.

2

payoffs are quite sensitive to the choices of others near the provision point. In such an

environment, even a small element of stochastic choice can have a significant effect on

incentives and decisions.

In order to examine the validity of this new explanation in a unified framework, I conduct

a two-by-two experiment; varying group size and information across treatments. While

previous experiments establish the puzzle, it is difficult to make clean comparisons across

heterogeneous parameterizations. For example, Croson and Marks (2000) demonstrate the

importance of the step return (SR), a threshold analog to the marginal per capita return

(MPCR) for continuous public goods (Isaac and Walker, 1988). It is not possible to identify

pure group size effects, without controlling for this variable. Throughout the experiment,

each participant’s endowment and the threshold are common knowledge. I am referring to

each player’s valuation of the public good, which are high or low with equal probability and

independently. In this context, complete information refers to knowing the realized binomial

state, whereas incomplete information refers to knowledge of the ex-ante distribution as well

as private knowledge of one’s own valuation. In order to prevent learning, valuations are re-

drawn every round.2 Furthermore, some information regarding the outcome of each round

is suppressed to control for distributional preferences, punishment or other common features

of repeated games. I measure the logit parameter in each treatment, in order to compare

predicted and observed provision rates.

As expected, I find that increased group size reduces success rates across both informa-

tion treatments. However, the effect is much smaller under incomplete information than

under complete information. For small groups, I also find that incomplete information re-

duces provision; although, the loss in efficiency is smaller than the effect of increased group

size within complete information. These results are compatible with the comparative statics

of the LQRE model. However, the experiment also reveals a strong “less-is-more” effect

for large groups, i.e., groups of eight are significantly better at provision with incomplete

2To my knowledge, this has not been done with provision point experiments. Marks and Croson (1999)either (i) tell subjects the aggregate valuation of the public good or (ii) do not give participants any infor-mation regarding others’ valuations.

3

information than with complete information. This counter-intuitive outcome is familiar

within the bounded rationality literature (Gigerenzer and Goldstein, 1996). Therefore, to

the extent that LQRE falls short in its predictions, the data are reinforcing the argument

that bounded rationality plays a key role in understanding the effect of information and

group size on outcomes.

The rest of the paper is organized as follows. In section two, I show that Nash equilibrium

predicts perfect provision, regardless of group size, and I show that the symmetric Bayes-

Nash equilibrium predicts a negative effect for incomplete information on provision. In

section three, I use the QRE model for complete information and the Agent QRE model

for incomplete information to predict provision rates as a function of the logit parameter.

Section four describes the experiment and the data, and presents results. Section five

concludes with discussion of possible extensions of the model for future research.

2 Bayes-Nash Equilibria

First, it is necessary to establish the negative effect of incomplete information on provision

rates.3 This is accomplished using the symmetric, pure-strategy Bayes-Nash equilibria to

make predictions. The intuition of the result is quite simple: provision will occur with

certainty in some states and will not occur with certainty in others; leading to an average

provision rate that is strictly less than one.

Early experiments on the provision point date back to Marwell and Ames (1980), who

included a provision point treatment in their series of experiments detailing the absence

of free-rider behavior in the laboratory. According to Kagel and Roth (1997), the pro-

vision point transforms the standard public goods game from a prisoner’s dilemma into

a game of chicken. A prisoner’s dilemma is characterized by it’s unique, inefficient Nash

equilibrium; while the game of chicken has two efficient equilibria; thus transforming the

3Looking at the Volunteer’s Dilemma, Weesie (1994) demonstrates a less-is-more Nash prediction, bycomparing mixed-strategy equilibria under complete information to pure-strategy equilibria under incompleteinformation. As stated earlier, Goeree et al. (2005a) reject the mixed-strategy explanation for group sizeeffects. Therefore, I focus exclusively on pure-strategy equilibria.

4

free-rider problem into a coordination problem. Subsequent experiments examined how

different parameterizations might effect this coordination.4

Bagnoli and Lipman (1989) establish the potential efficiency of the provision point mech-

anism under specific equilibrium refinements, for both unitary and multiple-unit public

goods. A follow-up experiment by Bagnoli and McKee (1991) finds some support for the

refinement in the unitary case, while Bagnoli et al. (1992) find strong evidence rejecting

predicted behavior in the multiple-unit case. The authors also choose to use proportional

refunds, which implies that efficiency and provision are equivalent, without creating per-

verse incentives for an individual to contribute more than her valuation.5 Later research

explored many extensions of the game, such as alternate rebate rules, or a first-mover that

chooses the threshold and receives the contributions, and whether players had imperfect

information regarding this first move.

2.1 The Model

The basic structure of the model is as follows. There are n players, indexed as i ∈ 1, ..., n.

Each player has a utility function defined over the public good and private consumption

ui(d,wi − ci) where d ∈ 0, 1 represents the unitary public good, wi ∈ R+ is player i’s

income, and ci ∈ [0, wi] is player i’s private contribution to the public good. I define each

player’s valuation of the public good as the amount vi that solves ui(0, wi) = ui(1, wi− vi).

If∑

i ci < T , then the public good is not produced (d = 0), all contributions are exactly

refunded, and each agent receives ui(0, wi). If∑

i ci ≥ T , the public good is produced

(d = 1), and the total excess contribution,∑

i ci − T , is refunded in proportion to each

4Early research focused on the role of the marginal per capita return (Isaac and Walker, 1988). Thiswas mainly manipulated by varying the threshold, in studies by Isaac et al. (1988), Rapaport and Suleiman(1992), Rapaport and Suleiman (1993), Palfrey and Rosenthal (1988), and Palfrey and Rosenthal (1991),with mixed results. However, Croson and Marks (2000) established the step return, SR, as the correctanalogy to the MPCR.

5The effects of different rebate rules spawned a separate research program. For example, Marks andCroson (1998) find that the presence of rebates affects the variance of contributions. However, I will simplyuse the same proportional refund rule, since I also use provision rates to compare the welfare predictionsbetween the Nash equilibria and the quantal response equilibria.

5

member’s contribution. Payoffs are normalized such that:

ui(0, wi) = 0 (1)

ui(1, wi − ci) = vi − ci +ci

ci +∑

j 6=i cj

ci +∑j 6=i

cj − T

(2)

Since each player’s payoff only depend upon the sum of others’ choices, the payoff function

may be more simply expressed using game-theoretic notation:

ui(ci, c−i) =

vi − T(

cici+c−i

)if ci + c−i ≥ T

0 if ci + c−i < Twhere c−i =

∑j 6=i

cj (3)

Appendix A.1 reviews the mathematical steps of Bagnoli and Lipman (1989) in this nota-

tion. The important result is that a strategy profile achieves∑

i ci = T if and only if it is

an undominated, perfect Nash equilibrium. Under this refinement, provision should occur

with probability one, so long as∑

i vi ≥ T .

2.2 Incomplete Information

A common intuition is that relaxing a complete information assumption will reduce welfare.6

In this section, I show that this intuition holds for pure-strategy equilibria in threshold

public goods games. To understand why, recall that efficient outcomes need undominated

strategy profiles. With incomplete information, a strategy profile is undominated if and

only if it is undominated in every state. Therefore, there will be equilibria which are only

dominated in some states that lead to a provision rate less than one. In order to solve

for these Bayes-Nash equilibria, it is first necessary to choose a parameterization of the

distribution of values. I define this distribution to be a Bernoulli random variable for

6As mentioned above, Weesie (1994) demonstrates why this need not always be the case with mixed-strategies.

6

parsimony and computational feasibility.

vi =

vl with probability p

vh with probability (1− p)(4)

The remaining parameterization of the model will hold throughout the rest of the paper,

as well as the experiment. I chose a low value of vl = 20, a high value of vh = 40. The

threshold to be T = nvl in order to control for step return effects. It also guarantees

that aggregate benefit is at least as high as the threshold in every state. Endowments

are equalized such that to wi = 40 simplify information . This means that only a low

type can bid more than her value. The probability of being a low type is p = 0.5. In

this environment, provision is efficient in every state. The approximated equilibria are

expressed algebraically in Figure 1 and visually in Figure 2. The undominated refinement

will only eliminate the equilibrium (0, 0). The remaining equilibria have a success rate

strictly between 0 and 0.6875. In contrast to complete information, equilibrium selection

will now determine the exact provision rate. Additional mathematical details regarding the

environment and methods of solving for these equilibria are found in Appendices A.2 and

A.3.

Figure 1: Bayes-Nash Equilibria

(a) Groups of 4∣∣ cl = 0 and ch = 0 (provision rate 0%)

(cl, ch)∣∣ cl = 0 and ch = 20 (provision rate 6.25%)∣∣ cl + 3ch = 80 for 0 < cl < 13.0 (provision rate 31.25%)∣∣ 2cl + 2ch = 80 for 13 ≤ cl ≤ 18.4 (provision rate 68.75%)

(5)

(b) Groups of 8

∣∣ cl = 0 and ch = 0 (provision rate 0%)∣∣ cl = 0 and ch = 20 (provision rate 0.39%)(cl, ch)

∣∣ cl + 7ch = 160 for 0.4 ≤ cl ≤ 7.4 (provision rate 3.52%)∣∣ 2cl + 6ch = 160 for 0.2 ≤ cl ≤ 11.9 (provision rate 14.45%)∣∣ 3cl + 5ch = 160 for 8.5 ≤ cl ≤ 15.5 (provision rate 36.33%)

(6)

7

Figure 2: Bayes-Nash Equilibria

(a) Groups of 4

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Low Contribution

Hig

h C

ontr

ibut

ion

(b) Groups of 8

0 5 10 15 20 25 30 35 400

5

10

15

20

25

30

35

40

Low Contribution

Hig

h C

ontr

ibut

ion

3 Quantal Response Equilibria

Bounded rationality is a highly interdisciplinary topic; especially between economics and

psychology. As such, there is significant variance in the formal definitions of bounded

rationality. The economics literature has tended two focus on two classes of models: (i)

Level-K models, which relax the equilibrium requirement of mutual consistency and (ii)

QRE models, which relax the requirement of strict optimization. These models are quite

natural extensions of existing game theory, so they are used most often. In fact, they have

even been combined into a larger, hybrid models such as heterogeneous QRE, or truncated

QRE which nest cognitive hierarchies as a special case (Rogers et al., 2009). 7

The stochastic nature of the quantal response framework lends itself naturally to the

issue of coordination. All of the equilibria I focus on are efficient, so equilibrium selection

only affects the distribution of benefits. Thus, someone always has incentive to change the

equilibrium selection in her favor. This creates noisy behavior. The quantal response equi-

7One notable exception to these approaches is the research on adaptive heuristics (Hertwig et al., 2011).This research is often critical of Level-K, cognitive hierarchies, and quantal response because such theoriesforce agents to make errors, while heuristic models of decision making are ecologically adaptive (Gigerenzerand Goldstein, 1996; Gigerenzer and Hoffrage, 1995; Gigerenzer and Brighton, 2009; Gigerenzer and Hof-frage, 1995). However, the model of noisy learning (Anderson et al., 2004) is also motivated by adaptivelearning. Furthermore, there are neighborhoods around the logit parameter for which increased rationalityis detrimental. Thus, it is not obvious that the standard models are incapable of satisfying the requirementsof the adaptive heuristics literature.

8

librium was first proposed by McKelvey and Palfrey (1995), including the logit specification

(LQRE), as an attempt to import statistical models of quantal choice into a game theoretic

model. Unfortunately, it is too general for testable restrictions on the data, and addi-

tional restrictions are required to have a Regular QRE (Goeree et al., 2005b). For those

who find the assumption that players do not strictly maximize unsatisfying, the authors

also prove the existence of Structural QRE, which demonstrates the isomorphism between

quantal response functions and utility maximization that is misperceived with an additive

error. Finally, the authors demonstrate that the logit specification satisfies the regularity

conditions, and may be defined structurally when the additive error to utility follows a

log-Weibull distribution.

To start, a continuous quantal response equilibrium is defined as:

i. A set of quantal response functions σ∗i ni=1 such that

σ∗i (ci) = fi (πi(ci)) (7)

where fi(·) is monotonic over πi(·) and a pdf over ci

ii. Beliefs are formed as the rational expectations to the quantal response functions of

others and determine the expected payoff, πi(ci) = ui(ci, σ∗−i), to each player

πi(ci) =

∫c−i

ui(ci, c−i) · σ∗−i(c−i) dc−i (8)

Logit equilibrium assumes a logit specification for the quantal response functions:

fi(πi(ci); λ) =exp(λ πi(ci))∫

siexp(λ πi(si)) dsi

(9)

For λ = 0 players randomize uniformly, and as λ −→ ∞, the quantal response equilibrium

selects a unique, pure strategy Nash equilibrium. It is important to note that this functional

9

form implies scale invariance of a sort:

fi(πi(ci); λ) = fi(kπi(ci); λ/k) (10)

This means that λ, the parameter denoting the degree of bounded rationality, does not

denote an absolute level of bounded rationality across all games. Rather, λ is a relative

measure that may not be comparable across games.

Using the quantal response functions, it is possible to construct the distribution of

total contributions and measure the fraction of the time that total contributions exceed the

threshold. This is the provision rate predicted by QRE. This requires solving the model in

every state for each treatment. To demonstrate the behavior of the model in a simple case,

first consider groups of two, in the state where one is a high type and one is a low type.

Figure 3 is an animation of the logit correspondence in this environment for λ ∈ [0, 0.5] under

complete information, while Figure 4 uses these same parameters, but extended to Agent

QRE (McKelvey and Palfrey, 1998) with incomplete information. Notice that predicted

behavior is not the same between information treatments, with respect to the provision

rate.

I solve the model for several group sizes n ∈ 2, 4, 8. Figure 5 plots the provision rates,

in the state where half are high types or low types. Interestingly, QRE predicts that infor-

mation does not matter for small values of λ, while group size does. As λ increases, and

best responses are less noisy, complete information benefits players. This makes sense, since

less boundedly rational players are able to exploit the additional information. However, the

higher the group size n, the higher λ must be for information to begin to matter. This sug-

gests that the strategic uncertainty regarding the play of others may dominate information

effects, when the number of players is large and players are sufficiently boundedly rational.

In order to make predictions, I calculate the average provision rate across all states; using

the frequencies realized in the experiment.

10

Figure 3: Groups of 2, Complete Information, λ ∈ [0, 0.5]

(a) Quantal Response Functions (b) Total Contributions

Figure 4: Groups of 2, Incomplete Information, λ ∈ [0, 0.5]

(a) Quantal Response Functions (b) Total Contributions

11

Figure 5: Provision Rates (Realization: half high/low)

0.0 0.2 0.4 0.6 0.8 1.0Lambda

0.0

0.2

0.4

0.6

0.8

1.0

Provision Rate

n=2 Completen=2 Incompleten=4 Completen=4 Incompleten=8 Completen=8 Incomplete

12

4 The Experiment

The experiment follows a two-by-two approach design, with large vs. small groups and

complete vs. incomplete information. Participants play the threshold public goods game

repeatedly across multiple unpaid and potentially paid rounds. Group assignments remain

static within the paid rounds, so that players have a chance to solve the coordination prob-

lem. However, players were assigned anonymously and they were not be able to communi-

cate, so as to minimize distributional preferences, reputational considerations, or learning.

The public good is described in terms of a monetary prize to avoid priming social norms

or altruistic motives common to public goods. I perform maximum likelihood estimates on

each treatment, to estimate the LQRE parameter. Theoretical predictions of the provision

rate, using the estimated logit parameter, are compared with actual provision rates to mea-

sure the performance of logistic quantal response in comparison to Nash and Bayes-Nash

equilibria. This is a genuinely falsifiable test of the model’s ability to explain the puzzle.

4.1 Design

Two sessions were run for each treatment. A small group had n = 4 while the large

group had n = 8. While Bagnoli and McKee (1991) and Marks and Croson (1999) use

n = 5, I desire an even number so I could set p = 0.5 and have the special state, half

high and half low, in which expectations are borne out exactly. Types were re-drawn each

round, regardless of which informational treatment was used. Under complete information,

each player knew the exact number of high and low types. However, under incomplete

information, each player knew her own type as well as the distribution from which others

types’ were drawn. Each player’s income was set to w = 40 for the entirety of the game.

High types valued the public good at vh = 40 while low types had vl = 20. The threshold

was set such that T = nvl.

The game was repeated many times. First, players experienced ten unpaid rounds, fol-

lowed by 20 potentially paid. Each observation is a given player in a given round. However,

13

as a robustness check, I only use the latter ten rounds to estimate the logit parameter. This

is consistent with the motivation of QRE as the steady state distributions of a dynamic

process of noisy directional learning (Anderson et al., 2004). At the end of the entire ses-

sion, one round was chosen at random to determine payoffs. Each participant received a $5

show-up fee. In addition, they received a $5 bonus, regardless of whether the threshold was

reached. This bonus was used to penalize over-bidding by low types, and could only be lost

if a participant contributed more than her valuation in the randomly chosen round. All of

this was clearly communicated to each participant, and all questions answered, before the

rounds began.

4.2 Maximum Likelihood Estimation

The logit quantal response correspondence makes a stochastic predictions for each value

of λ. This amounts to a prediction about the entire data generating process, and leads to

maximum likelihood estimation. First, it is straightforward to write down the likelihood of

any individual observation using the appropriate logit response function:

L(λ; ci) = σ∗i (ci; λ) (11)

The implicit maximum likelihood estimator for any given data set is:

λmle = maxλ

1

N

n∑i=1

ln (σ∗i (ci; λ))

(12)

s2mle =

[1

N

n∑i=1

(∂ ln (σ∗i (ci; λ))

∂λ λ=λ

)2]−1

(13)

By the central limit theorem:

√N(λmle − λ0)

smle

d−→ N(0, 1) (14)

14

Theoretically, this result only holds for interior solutions, i.e., for λ0 > 0. Moreover, the

logit model’s single parameter, λ, also determines the variance of contributions. This means

that the variance of λmle depends upon the null hypothesis, λ0. For these reasons, I conduct

simulations to verify standard errors for hypothesis testing.

4.3 Data & Results

An observation is defined as a individual player in a single round. For the four person, in-

complete information treatment, there was one experimental session with fewer participants.

In this case, the sample size was N = 560, while all other treatments had a larger sample

of N = 640. I use the full data set, since restricting attention to the latter rounds yields

similar predictions (see Appendix C). One pronounced characteristic of the data is that

most participants chose whole dollar contributions. In Figure 6, multiples of five are most

prominent in the data; representing over half of all choices. Histograms of the data that

have sufficiently wide bins will mask this anomaly. Using the quantal response functions

at the estimated parameter values, I integrate over these densities to produce a histogram

of the model that is comparable with a histogram of the data. Figure 10 compares these

histograms of the quantal response functions with histograms of the data, using wide bins.

There are a considerable number of observations in which low types over-bid, i.e., they

contribute more than $20, despite explicit instructions and verbal explanations of how

one could lose money. It is not obvious that this over-contribution represents bounded

rationality, as it could represent some other phenomena such as implicit reciprocity between

subjects across rounds. However, it is clear that the QRE predicts the frequency of over-

contribution quite well, as seen in Figure 7.

Of course, the most important analysis of the data involves the maximum likelihood

estimator admitted by the quantal response model. I use the data from all twenty paid

rounds. For incomplete information, this is simply a matter of using the two quantal

response functions (one for each type) to construct the likelihood function. However, the

complete information case is a bit more complex. In the experiment, values are re-drawn

15

Figure 6: Histogram of Contributions, by type, with fine bins

0.00

0.05

0.10

0.15

0.20

0.25

0.30

Probability

0 5 10 15 20 25 30 35 40 45Contribution

Figure 7: Contributions Above Valuation

(a) Predicted by Model

Complete Incomplete

n = 4 0.156 0.147n = 8 0.190 0.202

(b) Observed in Experiment

Complete Incomplete

n = 4 0.144 0.141n = 8 0.133 0.133

every single round, so each round may have a different number of high types and low

types. Under complete information, quantal response functions are state-specific. Thus,

for example, there are eight quantal response functions used in the maximum likelihood

estimation of the four-person, complete information treatment. Below are the results for

each treatment.

First, it is immediately apparent from Figure 8 that 0 < λ < 1 for every treatment.

Given the standard errors, it is clear that λ0 = 0 and λ0 = 1 are not close to any confi-

dence intervals, so I reject that players choose uniformly, and I reject whichever underlying

Nash equilibrium the limiting logit selects. The only two values of lambda that are not

16

Figure 8: Maximum Likelihood Estimates (all 20 rounds)

(a) λmle and semle

Complete Incomplete

n = 4 0.320(0.014)

0.523(0.030)

n = 8 0.440(0.027)

0.557(0.045)

significantly different are the two incomplete information treatments. However, restricted

attention to the latter ten paid rounds (as in Appendix C), these estimates are statisti-

cally different. More importantly, the estimates of λ have very little interpretation across

treatments and group size, as these transformations to the payoff matrix are beyond the

invariance condition. The most appropriate measure is the predicted provision rates, which

are calculated state-by-state and then averaged across states. However, unlike before, I will

now use the ex-post frequencies of each state, rather than the ex-ante binomial probabilities.

That way, any gap between the model’s predictions and the observed rates does not reflect

a gap between the ex-ante and realized distribution of types.

Figure 9: Average Provision Rates (all 20 rounds)

(a) Model Prediction at λmle

Complete Incomplete

n = 4 0.405 0.320n = 8 0.180 0.174

(b) Observed in Experiment

Complete Incomplete

n = 4 0.675(.0185)

0.586(.0208)

n = 8 0.425(.0196)

0.525(.0198)

It is clear from Figure 9 that the QRE systematically under-predicts the average pro-

vision rates in all treatments. Still, it does an acceptable job rank ordering provision rates

across treatments. Among small groups, incomplete information decreases provision, but by

less than increasing group size within complete information. While these comparative stat-

ics are accurate, there is one notable exception. The observed average success rate jumps up

ten percentage points, for groups of eight, when going from complete to incomplete infor-

mation. This is an example of the so-called “less-is-more” effect (Gigerenzer and Goldstein,

17

1996). It is equally apparent when restricting attention to the latter ten rounds. While it is

technically possible for the quantal response model to predict the direction of this effect, it

would require either a value of λ ∈ [0, 0.2]. The higher success rate would be driven purely

through extra noise and over-bidding, due to much flatter and noisier distributions.

Figure 10: Groups of 8 / Incomplete Information

(a) Prediction at λmle

0.0

0.1

0.2

0.3

0.4

0.5

Probability

2 6 10 14 18 22 26 30 34 38Contribution

(b) Experimental Data

0.0

0.1

0.2

0.3

0.4

0.5

Probability

2 6 10 14 18 22 26 30 34 38Contribution

Glancing at Figure 10, the quantal response model seems to do a decent job of matching

the data. There is enough difference in the distributions to suggest that something is

missing. For example, the peaks seem to be (i) higher and (ii) closer together. While

increasing λ would raise the peaks of the distribution, it will not bring them closer together,

and it will lower the provision rate. It is also clear from these histograms that lowering

λ to match the provision rate would be a mistake. Flattening the model’s distributions

further would help to match that moment, but it would move the model as a whole farther

away from the data. The Kolmogorov-Smirnov test provides rigorous evidence rejecting the

quantal response functions as the data generating process. However, that may be due to the

concentration of observations at integer values. I also conduct the chi-square goodness-of-fit

test using the histograms in Figure 10. This also leads to rejection of the quantal response

functions in every treatment. These results can be found in Figure 11. Therefore, the model

must be extended, in order to fully explain the data.

18

Figure 11: Data generating process tests

(a) Kolmogorov-Smirnov Low Types

Complete Incomplete

n = 4 KS = .3827p=.0000

KS = .4920p=.0000

n = 8 KS = .3648p=.0000

KS = .3734p=.0000

(b) Chi-Square Low Types

Complete Incomplete

n = 4 χ2 = 138.81p=.0000

χ2 = 295.91p=.0000

n = 8 χ2 = 72.27p=.0000

χ2 = 220.35p=.0000

(c) Kolmogorov-Smirnov High Types

Complete Incomplete

n = 4 KS = .1648p=.0001

KS = .2030p=.0000

n = 8 KS = .2274p=.0000

KS = .2595p=.0000

(d) Chi-Square High Types

Complete Incomplete

n = 4 χ2 = 83.69p=.0000

χ2 = 113.05p=.0000

n = 8 χ2 = 37.59p=.0000

χ2 = 121.26p=.0000

5 Conclusion

This paper examines a puzzle regarding the provision point mechanism. Namely, theory

is silent about equilibrium selection, while experiments suggest this coordination problem

is harder for large groups. Likewise, standard theory predicts significant, negative welfare

effects for incomplete information, but this is not borne out in all experimental data. Using

the logit specification of quantal response equilibrium, I demonstrate how bounded ratio-

nality may resolve this puzzle. In this framework, large groups will experience negligible

information effects near the estimated parameter values. Furthermore, large groups actu-

ally demonstrate a less-is-more effect in the experiment; whereby less information leads to

an economically and statistically significant increase in welfare. This effect is robust when

restricting attention to the latter ten rounds.

This research is an important step in understanding applications of the provision point

mechanism such as www.kickstarter.com and www.indiegogo.com. The growing popularity

of such online crowd funding sites is transforming the traditional environment of threshold

fund raising away from small groups or local communities, and toward disparate groups of

thousands or millions of strangers spread over a wide geographic and demographic range.

Models that resolve the puzzle presented here, may also shed light on the effectiveness of

crowd funding in such large groups with so little information.

19

Similar economic settings that receive lots of attention, such as market entry games,

may also benefit from this research. For example, market entry games are characterized by

a threshold, representing the maximum number of firms that can enter a market profitably.

Many equilibria exist, since many subsets of firms can enter the market without exceeding

the capacity. Private information now refers to cost of entry, rather than benefit from the

public good. Noisy decision-making with respect to market entry, with lots of firms and

little information about competitors’ costs, may explain the observed excess entry in an

equilibrium model with correct beliefs, rather than overconfidence (Camerer and Lovallo,

1999). This research also contributes to the work on more general issues such as equilibrium

selection and coordination.

The fact that the experiment yields a less-is-more effect bolsters the argument that

bounded rationality is playing a large role in information effects. Selecting a lower value

for λ in order to match this effect would generate quantal response functions that are less

similar to the data. Extending of the standard QRE model may be sufficient to capture

this remaining effect. There are several obvious choices, such as heterogeneous QRE or

truncated QRE (Rogers et al., 2009), representative agent QRE (Golman, 2011), or QRE

with altruism (Anderson et al., 1998). Additionally, it would be worthwhile to re-run the

experiment without the rebate of excess contributions. This would be a natural test to

see if the observed less-is-more effect is simply tacit cooperation. This would also yield

an additional QRE comparative static to compare with the literature, such as Marks and

Croson (1998) who find the rebate rules affect the second moment of contributions. This

paper presents just one example of how QRE models can be used to resolve interesting

puzzles that challenge common economic intuition.

20

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24

A Nash and Bayes-Nash Equilibria

A.1 Complete Information: General Model

Each player’s best response is to pay the minimal amount necessary to generate the public

good given the contributions of every other player. However, no player can bid less than

zero, and individual rationality requires that no player would bid more than her valuation if

the public good is produced. Unfortunately, this framework is not sufficient to guarantee the

efficiency of Nash equilibria. For a stark example, consider n = 2, v1 < v2 < T < v1 + v2.

Then one Nash equilibrium is (c1, c2) = (0, 0), since neither player can best respond by

unilaterally funding the public good without violating individual rationality. This basic

problem can take a much more general form.

Consider any game such that vi < T for all i and∑

i vi > T . Then any strategy

profile (c1, ..., cn), such that ci + c−i < T and vi + c−i < T holds for every player i, is

an inefficient Nash equilibrium. The problem is that any individual player who increases

her contribution to by the amount necessary for provision will force to violate individual

rationality. To address this problem, it is necessary to use the refinement of undominated

Nash equilibria:

CU (v1, ..., vn) =

(c1, ..., cn)∣∣ vi + c−i ≥ T for at least one i

(15)

For example, consider the two person game from above. Given complete information and

individual rationality, player 1 knows that player 2 will never choose a contribution c2 > v2

when d = 1. However, if he chooses to contribute v1 ∈ [0, T − v2), he is unilaterally

guaranteeing that the public good is not produced. Likewise, this is true for player 2 and

v2 ∈ [0, T − v1). The undominated Nash equilibrium refinement to restricts attention to a

subset of the strategy space: (c1, c2) ∈ [T − v2, v1]× [T − v1, v2]. Thus, senseless equilibria

such as (c1, c2) = (0, 0) are ruled out.

Unfortunately, this is only one dimension of the problem. In fact, I can construct other

25

disturbing examples. Suppose n = 3, v1 < v2 < v3 < T < v1 + v2 + v3, and further suppose

that v2 +v3 < T . Let ε ∈ (0, T −v2−v3). Then the strategy profile (c1, c2, c3) = (0, 0, v3 +ε)

is a set of inefficient Nash equilibria in which player 3 is violating individual rationality.

More generally, let ε > 0 and suppose vi + c−i + ε < T . Then player i is indifferent between

all strategies ci ∈ [0, vi + ε]. These equilibria are eliminated with a stability condition.

Consider trembling hand perfect equilibria, where there is a small chance that, say, d = 1.

Then player i strictly prefers to play any ci ∈ [0, vi] to any ci ∈ (vi, vi + ε].

So if I restrict attention to perfect equilibria over undominated strategies, does that

guarantee efficient outcomes? No. It is not enough that player’s choose undominated

strategies in perfect equilibrium. Rather, they must also choose undominated strategies

as trembles, else I might still have inefficient, implausible outcomes. To see this, consider

n = 2, v1 = v2 = 0.6, T = 1. Let’s start from the strategy profile (c1, c2) = (0, 0). Suppose

there is a small probability that player 2 will accidentally choose c2 = 1, thereby generating

the public good. Then the first player’s best response is still c1 = 0; strictly so if c2 = 1.

Notice that this exotic exception requires that player 2 tremble by contributing more than

his valuation. Therefore, if I restrict attention to undominated perfect equilbria, meaning

that trembles must also be undominated, then the remaining equilibria will be efficient and

stable such that∑

i ci = T .

A.2 Incomplete Information: General Model

Of course, everything mentioned so far implicitly assumes certainty. Suppose I generalize

the model to include states of nature ω ∈ Ω. For the sake of focus, I shall simply assume that

wi(ω) = wi ∀ ω. More interestingly, I will consider state dependent utility ui(d,wi − ci| ω)

in order to have state-dependent willingness-to-pay vi(ω) ∈ Vi(Ω). In this framework,

the previous case of complete information is equivalent to knowing the state of the world(v1(ω), ..., vn(ω)

). In order to relax this assumption, I will assume that each player i knows

her own valuation vi with certainty and has correct beliefs over V−i(Ω).

By analogy to complete information, I will examine undominated, trembling hand per-

26

fect Bayes-Nash equilibria. By assuming trembling hand perfection, I shall not consider

equilibria for which ci > vi. Therefore, there are only two differences in equilibrium

between complete and incomplete information. The incomplete information over other

players’ valuations creates uncertainty about which set of undominated strategies to use.

To see this, recall that the set of undominated strategies is a function of all valuations

CU = CU(v1(ω), ..., vn(ω)

). For simplicity, I shall write CU = CU (ω).

If a strategy is dominated in some states, but not others, than the expected payoff

will be positive. Thus, it will not be a dominated strategy with respect to expected util-

ity. Therefore, I must weaken my definition of undominated strategies to those that are

undominated in at least one state of nature, i.e.,

CUi (Ω) =⋃ω

CUi (ω) (16)

For example, consider n = 2, vl < vh < 2vl = T , and further suppose there are four

states of nature(v1(ω), v2(ω)

)∈ (vl, vl), (vl, vh), (vh, vl), (vh, vh), each with probability

1/4. Denote these states as ω1, ω2, ω3, and ω4 respectively. In state ω1 all strategies ci ∈

[0, vi] are weakly dominant. In this case, the undominated refinement has no bite. To

advance the discussion further, I shall now consider a simple parameterization, where player

types are distributed i.i.d. Bernoulli.

A.3 Incomplete Information: Bernoulli Types

In order to solve for and characterize some Nash equilibria, I shall choose a simple distri-

bution over types for players. Specifically, assume that each player’s type is a Bernoulli

random variable:

vi =

vl with probability p

vh with probability 1− p(17)

Of course, players know their own type. So, with n − 1 other players there are still 2n−1

states of nature. However, it is combinations rather than permutations that will effect

27

players decisions. So, I may consider the states of nature, Ω to be summarized by the

number of players that are low types, ω ∈ 0, 1, ..., n− 1. Therefore, ω follows a binomial

distribution:

P (ω) =

n− 1

ω

pω(1− p)n−1−ω (18)

To further simplify the model, notice that players of the same type all face the same

problem. Therefore, I shall focus my attention on equilibria (c∗1, ..., c∗n) such that:

c∗i =

c∗l if vi = vl

c∗h if vi = vh

(19)

This simplification will make equilibria two dimensional, (c∗l , c∗h). Imposing the symmetry

condition, I define the payoff function as:

ui(ci, cl, ch| ω) =

vi − Tcici+ωcl+(n−1−ω)ch

if ci + ωcl + (n− 1− ω)ch ≥ T

0 if ci + ωcl + (n− 1− ω)ch < T(20)

The expected payoff function is:

πi(ci, cl, ch) =

n−1∑ω=0

n− 1

ω

pω(1− p)n−1−ω · ui(ci, cl, ch| ω) (21)

Then I perform a grid search for the set of symmetric Bayes-Nash equilibria:

(c∗l , c∗h) =

(cl, ch)

∣∣∣ c∗i = cl if vi = vl and c∗i = ch if vi = vh

(22)

B Logit Quantal Response Equilibria

B.1 Structural QRE with Noisy Learning

Anderson et al. (2004) demonstrate that every potential game has a unique and globally

28

stable continuous Logit equilibrium. One necessary and suffient condition for a potential

game is that payoffs are of the form:

ui(ci, c−i) = u(ci, c−i) + θi(ci) + φi(c−i) (23)

where u(·) is common across players, but θi(·) and φi(·) are specific to each player. It is

straightforward to see that the threshold game satisfies this condition:

ui(ci, c−i) = vi︸︷︷︸θi(ci)

−T(

cici + c−i

)︸ ︷︷ ︸

u(ci,c−i)

+ 0︸︷︷︸φi(c−i)

(24)

Therefore, a continuous analog exists to the principal branch of the Logit Quantal Response

Equilibrium, which can be thought of as the steady state to a dynamic process of noisy

learning with normally distributed error.

B.2 Complete Information

Recall that players’ utility functions only depend on the sum of others’ choices:

ui(ci, c−i) =

vi − T(

cici+c−i

)for ci + c−i ≥ T

0 for ci + c−i < T(25)

Thus, the logit equilibrium is a system of functional equations:

πi(ci) =

∫c−i

ui(ci, c−i) dFi(c−i) (26)

The expected payoffs functions will be continuously differentiable, implying continuously

differentiable logit response functions. To see this, note that I may express the bounds of

integration so as to ignore the discontinuity of ui(·):

πi(ci) =

∫c−i≥T−ci

[vi − T

(ci

ci + c−i

)]dFi(c−i) (27)

29

I assume an environment with two types (vl, vh). I further assume a symmetric equilib-

rium with respect to type. Therefore, the random variable c−i generated by the indepen-

dent draws from others’ logit response functions will only admit two distribution functions

(Fl, Fh). Thus, I want to solve for two functions (πl, πh) such that:

πl(ci) =

∫c−i≥T−ci

[vl − T ·

cici + c−i

]dFl(c−i) (28)

πh(ci) =

∫c−i≥T−ci

[vh − T ·

cici + c−i

]dFh(c−i) (29)

Note that the Logit transformation also implies the following:

∂fi(πi(ci))

∂ci= λfi(πi(ci)) ·

∂πi(ci)

∂ci(30)

Using Leibniz’s integral rule, I find the following:

∂πl(ci)

∂ci= vl − ci +

∫c−i≥T−ci

[vl − T ·

c−i(ci + c−i)2

]dFl(c−i) (31)

∂πh(ci)

∂ci= vh − ci +

∫c−i≥T−ci

[vh − T ·

c−i(ci + c−i)2

]dFh(c−i) (32)

Thus I ar left with the following pair of differential equations:

∂fl(πl(ci))

∂ci= λfl(πl(ci)) ·

[vl − ci +

∫c−i≥T−ci

(vl − T

c−i(ci + c−i)2

)dFl(c−i)

](33)

∂fh(πh(ci))

∂ci= λfh(πh(ci)) ·

[vh − ci +

∫c−i≥T−ci

(vh − T

c−i(ci + c−i)2

)dFh(c−i)

](34)

These equations imply the solution to a quantal response equilibrium, whose existence

was proven in McKelvey and Palfrey (1995). The algorithm is a discrete approximation

exploiting the homotopy equations in Turocy (2005).

30

B.3 Incomplete Information

Now, I must further generalize the notion of expected utility. Before, I were actually

considering the expected utility state-by-state, where πωi (ci) is the expected payoff generated

by the logit response functions of others in a given state of nature. Now, expected utility

will be:

E[πωi (ci)] =

∫ω∈Ω

[∫c−i

ui(ci, c−i) · Fi(c−i) dc−i

]dPi(ω) (35)

Extending the logit equilibrium to the incomplete information case is theoretically

straight forward. Given the logit response functions by type, I can simulate the total

contributions of others state-by-state:

E[πωl (ci)] =∑ω∈Ω

Pl(ω)

∫c−i

ul(ci, c−i) dFωl (c−i) (36)

E[πωh (ci)] =∑ω∈Ω

Ph(ω)

∫c−i

uh(ci, c−i) dFωh (c−i) (37)

Recognizing that the distribution of types is binomial, and restricting the bounds of inte-

gration, I can express the problem as follows:

E[πωl (ci)] =n−1∑ω=0

n− 1

ω

· pω(1− p)n−1−ω∫c−i≥T−ci

[vl − T ·

cici + c−i

]dFωl (c−i) (38)

E[πωh (ci)] =

n−1∑ω=0

n− 1

ω

· pω(1− p)n−1−ω∫c−i≥T−ci

[vh − T ·

cici + c−i

]dFωh (c−i) (39)

The equations above imply the solution to an Agent Quantal Response Equilibrium, whose

definition and existence were proven by McKelvey and Palfrey (1998). The algorithm relies

on the discrete approximation using homotopy methods described in Turocy (2010).

C Data, Estimation, and Simulation

31

Figure 12: Maximum Likelihood Estimates (last 10 rounds)

(a) λmle and semle

Complete Incomplete

n = 4 0.330(0.029)

0.459(0.040)

n = 8 0.450(0.042)

0.657(0.071)

Figure 13: Average Provision Rates (last 10 rounds)

(a) Model Prediction at λmle

Complete Incomplete

n = 4 0.410 0.322n = 8 0.180 0.166

(b) Observed in Experiment

Complete Incomplete

n = 4 0.722(.0251)

0.579(.0296)

n = 8 0.456(.0279)

0.566(.0278)

Figure 14: Groups of 4 / Incomplete Information

(a) Prediction at λmle

0.0

0.1

0.2

0.3

0.4

0.5

Probability

2 6 10 14 18 22 26 30 34 38Contribution

(b) Experimental Data

0.0

0.1

0.2

0.3

0.4

0.5

Probability

2 6 10 14 18 22 26 30 34 38Contribution

32