The admin sequence for each program ends with one or two "results" pages
that show all data and some summary average calculations. In addition, most
programs offer graphs that show the sequence of decision averages or
transactions prices.
The graphs can be customized, e.g. by showing or hiding
predictions and by showing data points one at a time in sequence (buttons
for doing this have been removed from the graphs on this sample data display page).
Please note: Unless otherwise noted, these are
classroom "experiments" in which one student was selected at random
ex post to be paid a small fraction of earnings, using the instructor's
own pocket money. These exercises were done for instructional purposes,
and results may differ from those of research experiments
with anonymity, real money payoffs, no expectation of subsequent class
discussions, etc. The submit buttons on these pages have been disabled
or removed; use the "Return to Data Display" links to return to this page.
- Bargaining:
Data Table for Two-Stage, Alternating Offer Game
Graph for Two-Stage Alternating Offer Game
Comment:
The links show data for a two-stage alternating-offer bargaining game,
first in a table that shows individual decisions and earnings, and then in a
graph of the average demands and counter-proposals. In treatment 1,
the subgame perfect Nash prediction is for (rational, selfish) proposers to demand about sixty percent of the pie (which
shrinks from $5 to $2 in the event of a rejection). Average proposals track
this prediction well. In treatment 2 with 90 percent shrinkage, the
proposers generally do not demand this much, and such high demands are
typically rejected.
- Bayes' Rule Probability Elicitation:
Data Graph
Comment: The graph shows results of a research experiment
at UVA with cash payments.
Participants observed draws from a cup of colored balls, with
the contents determined by the unobserved state. Participants reported a
probability for one of the states, and the graph shows the average of the elicited
probabilities for the "Red Cup," with the Bayes' Rule prediction on the
horizontal axis. The incentive mechanism was designed to induce truthful reporting,
regardless of risk attitudes. There is some variability, but decision averages (in blue) lie close to the
red 45-degree line implied by Bayes' rule.
- Bertrand Market:
Data Graph
Comment: Prices are relatively low and stable, even in treatment 1 with
fixed matchings, where only one of the 6 triopoly groups managed to achieve significant
tacit collusion, with prices above $9 for that group. Prices are declining and end up somewhat closer to
the Nash/Bertrand (marginal cost) level in treatment 2 with random matchings.
Note that the person who designed this classroom experiment changed both cost and matching method,
which makes the results a little more difficult to evaluate.
- Takeover Game:
Data Graph
Data Table for Takeover Game
Comment: The top link shows the average offers made by buyers
who did not know the value of the firm, but who did know that it would be
worth 1.5 times as much for them as for the seller. Seller costs are
randomly generated on an interval that changes in the second treatment.
The "buyer's curse" effect in the first treatment results in significant losses
for most bidders, who are bidding above the Nash prediction in that treatment.
- Call Market:
Graph of Call Market Results
Comment: Students were allowed to update bids and asks, which were continuously crossed
to determine a provisional uniform price, that became the final price at the end
of the 5 minute period. The value and cost parameters were randomly
drawn from the same distributions at the start of each round. The graph
shows the supply and demand (in blue) for the fifth period, along with the
bid and ask arrays (in gold and orange respectively). These bid and ask
arrays are essentially flat at a common price that is in the range
of the competitive price prediction. Efficiency reached 100% (of possible
earnings) in this round, although it was somewhat lower in the first
couple of rounds.
- Centipede Game:
Graph of Average Number of Stages Reached
Comment: The data are from a classroom experiment with 22 students who
were randomly matched in a six-stage centipede game. A person who stops in the first stage earns 40 cents,
and the other earns 10 cents. These payoffs are doubled in each successive stage.
The Nash prediction is shown as a blue horizontal line.
- Common Value Auction with random groupings:
Common Value Auction (Group Size = 2)
Comment: The common value in each auction was the average of the two
bidders' signals. Bids exceeded the Nash prediction of one-half of one's own signal value.
Negative earnings for some periods are shown in red. Overbidding due to
the "winner's curse" was more pervasive in a classroom experiment involving
a relatively large group:
Common Value Auction (Group Size = 12)
which is a nice way to set this up for class discussion.
- Coordination (Minimum Effort) Game:
Graph of Average Efforts
Comment: The reduction in effort cost raises efforts, even though any
common effort is a Nash equilibrium for both treatments.
- Cournot/Monopoly Markets:
Individual Results Page
Summary Graph
Comment: Outputs converge to the monopoly prediction (treatment 1) and to
the Cournot/Nash duopoly prediction (treatment2). The next graph shows a Cournot
duopoly with random matchings followed by fixed matchings. Outputs converge
to the Cournot/Nash prediction in each case, but the variability is much less
under fixed matching.
Graph: Fixed Versus Random Matchings
- Demographic Questionnaire:
View Survey Questions
Comment: Each program offers the option of having participants fill out
a short demographic questionnaire after they receive results for the final
round of the experiment. The questionnaire, which takes about 2-3 minutes
to complete, is designed to provide basic demographic information (gender,
race or ethnic background, income, education, etc.) that may be useful
for the analysis of behavior in research experiments. Class discussions may
focus on things like whether men are less risk averse or less likely to
contribute in a public goods game, for example. Normally, individual responses
to such surveys are confidential, and the researcher should be careful to
follow guidelines and obtain approval from the relevant human subjects committee
before using this Questionnaire in research.
- Double Auction:
Graph of Supply, Demand, and Price Sequences
Period 1 Aggregate Results and Efficiency
Period 1 Log of Bids and Asks
Buyer Decision Sheet
Seller Decision Sheet
Comment:
The top link shows a graph of the supply,
demand, and transactions price series for a classroom double auction done at Wesleyan University.
The final four links pertain to a software demonstration,
showing efficiency calculations, a "ticker tape" price log,
and individual decision sheets.
- Guessing Game:
Data Table with Individual Decisions
Graph of Average Predictions by Round
Comment: The first treatment consists of 5 rounds with 5 participants who chose
numbers between 0 and 100. The $10 prize went to the person whose number was
closest to one-half of the average of all numbers. The guesses are
above the Nash prediction (0) in the first round, and the person who used several
rounds of iterated strategic thinking and chose 6.25 was the winner. The data converge
to near-Nash levels by round 5. In the second treatment, there is also a
$10 prize for the person whose number is closest to a "target,"
but the target is 20 plus one half of the average.
The average guess converges to a level that is near the Nash prediction (40)
by the final round. The first time that I ran this setup in one of my own classes,
I had incorrectly assumed that the Nash equilibrium was 20, since this was 20 above
the Nash prediction for treatment 1. The students playing the game
were quick to learn to respond to other's decisions, and this learning caused decisions
to converge to the correct Nash prediction.
- Matrix Game (2x2):
Data Table
Comment: This classroom experiment began with 10 periods of a prisoner's
dilemma game with random matching. Cooperation rates fall to zero,
as predicted in a Nash equilibrium for a one-shot prisoner's dilemma. The
second treatment is a symmetric matching pennies game with random matching,
and the mixed strategy Nash prediction is for each person to choose each
decision with probability 1/2.
- Large (NxN) Matrix Game:
3x3 Matrix Game Results
Comment: This was for a classroom experiment run at Emory University. The
choices for Row and Column players are concentrated at the unique
Nash equilibrium (Row 2, Column 3), even though this does not maximize
the minimum payoff for each player.
- Lemons Market:
Graph of Transactions Prices and Quality Grades
Comment: The market consists of 5 sellers and 7 buyers. Sellers choose prices and quality grades
independently in each round, with a middle grade of 2 being optimal in
the sense of maximizing total surplus. When prices and grades are
observed by all, most units sold are of grade 2. In the final 6 rounds,
buyers can see price but not grade prior to purchase, and grades fall to
1, which results in a reduction in surplus by more than 50 percent.
- Limit-Order Asset Market:
Graph of Market-Clearing Prices
Comment: Participants were given endowments of cash and "shares" of an asset
that had an equal chance of paying dividends of $0.40 or $1.00 in each round.
Cash could be used to buy other shares, and cash retained at the end of each round
would earn 10 percent interest. It was announced that all shares held at
the end of round 40 would be redeemed for $7.00. This was a research experiment
with 12 traders, and payments were made on a 1/100 rate. Traders submitted limit orders to buy and/or sell, with the final clearing price
being determined by the intersection of the bid and ask arrays, as in a "call market."
Each vertical line in
the graph indicates the end of a round, and the blue lines show the market-clearing
prices, with the horizontal distance being the trading volume. The horizontal red line at the bottom
shows the present value of the asset.
- Lottery Choice Menu Used to Measure Risk Aversion:
High Real Payoff Lottery Choice: Individual Results
High Real Payoff Lottery Choice: Data Graph
Comment: The setup is one for which fewer than 4 safe (S) choices indicates risk
preference, exactly 4 safe choices indicates risk neutrality, and more then 4 safe choices
indicates risk aversion. This was a research experiment done at UVA with high earnings, up to
$77.00, and very high levels of risk aversion are observed. The thin black line in
the associated graph shows the risk neutrality prediction; the data averages
are well to the right of this line. The next link shows data for a comparable
one-round experiment with much lower payoffs (only up to $3.85).
Low Real Payoff Lottery Choice: Data Graph
Payoff scale effects are likely to be much lower in
classroom experiments with hypothetical earnings (or with one
person selected at random and paid a small fraction of earnings).
The following graph shows results of a
classroom experiment (with random ex post partial payments) in
which an increase in the payoff scale from round 2 to round 3
causes a similar, but smaller, increase in risk aversion.
Data for a Classroom Experiment with Essentially Hypothetical Payoffs
- Political Event Market - 2004 Presidential Vote Share Prices:
Graph of Vote Share Prices on October 21
Comment: About 50 participants were members of classes at UVA and William and Mary.
Participants could make deposits that could be used to buy $1 "market portfolios," each of which
consisted of one share of Bush, one of Kerry, one of Nader, and one of Rest of Field.
Each share for a particular candidate pays a penny amount that equals the percentage of the
popular vote received by that candidate in the election.
Traders submit limit orders to buy and/or sell. The bid/ask arrays are crossed
each day at midnight to determine a market clearing price for each candidate.
The share prices at this point indicated that the election is in a dead heat.
- Posted-Offer Auction:
Graph of Supply, Demand, and Transaction Prices
Comment: This was a classroom experiment with 5 sellers and 7 buyers.
Two "large" sellers had numerous units with costs that equaled the
competitive price prediction of $2.60. These sellers had "market power" to
raise price above the competitive level profitably, and this may have
been a factor in the fact that prices converged to about $2.80, although
efficiency was at 100 percent.
- Principal-Agent Game with Contract Choice:
Individual Results Table
Graph
Comment: This was a classroom experiment run in a Contract Law class
at the Virginia Law School. Each of the 12 employers began by choosing a
contract that specified a suggested effort
and a fixed wage to be paid regardless of the worker's effort.
Employers also chose between
an "incentive contract" with a potential punishment and a "bonus contract" with
a non-binding promise of an ex post bonus. The wage had to cover the
worker's cost of effort, and upper limits on possible penalty and bonus payments
were enforced. A penalty could only be imposed if low effort could be
verified by a third party, which occurred with a pre-specified probability
of 1/3. Workers could either reject the contract or
accept and select an effort from 1 to 10. About three-fourths of the contracts were bonus
contracts. Effort levels decline, and about half of the efforts end up
at the minimum level of 1. Efforts are lower for penalty contracts.
- Private Value Auction with random pairings:
Individual Results for a First-Price Auction
Graph for First-Price and Second-Price Auctions
Comment: The individual results page shows data for a first-price auction.
The average bid/value
ratios on the left suggest that there is no
time trend in the typical tendency to bid at a level that is slightly
above Nash predictions for risk-neutral bidders.
The graph link above shows data for a matched pair of
first-price and second-price (Vickrey) auctions with three-person groups;
bid/value ratios are close to Nash predictions. Finally, the
graph link below shows a typical pattern of bidding above value in
a large-group second-price auction.
Graph of Second-Price Auction
- Probability Matching:
Graph of Prediction Proportions
Comment: In the first 50 rounds, event 1 tended
to occur with probability 0.75, and this probability is reduced to 0.25 for the
final 50 rounds. Actual choice frequencies veer away from these "probability
matching" predictions in the direction of choosing the more likely event
every time, which is the rational decision. Only one of the 12 participants in
this classroom experiment was picked ex post to be paid. In contrast, psychologists
have observed probability matching behavior in experiments with hypothetical payoffs.
And animal subjects (e.g. rats) with food motivation appear to be more rational than
people with hypothetical payoffs.
- Provision-Point Public Goods Game
The graph shows results from a classroom experiment run at the USAF Academy with
100 participants, divided into groups of size 4. Participants were given 10 tokens in each round,
which could be kept and redeemed for $1 each, or contributed to a public good. If the total
number of tokens contributed by group members reached 25, then each person would receive $20 (plus the
value of tokens kept),
regardless of whether or not they contributed any tokens. Average contributions fell below
the provision point in the
first "no-rebate" treatment, in which contributions were not returned in the event that the
provision point was not reached. In contrast, average contributions were about equal to the
provision point in a second treatment in which contributions were returned if the
provision point was not reached.
Graph: Rebate Effect
- Public Goods Game
All students in this classroom experiment were paid their earnings (full earnings for the 3-person group
and half of earnings for the 5-person group):
Three-Person Group
Five-Person Group
Comment: The Average column shows the per-capita average contribution (out
of an endowment of 25 tokens). Contributions tend to decline, except for some
spikes caused by signaling. Contributions are considerably higher
with the larger group, although this strong numbers effect is not always
observed in games with high returns. (Both groups used an internal return (to oneself) of
.6 and an external return (to each of the others) of .8.) The graph that follows
shows results for an environmental economics class at Dickinson College where
the internal return is increased by a factor of 4, which cuts the private cost
of contributing by a factor of 4. This payoff change has no effect on
the Nash (free-riding) prediction under the assumption that people are
selfish, but it causes contributions
to increase, especially in the final two rounds.
Graph: Change in the Internal Return
- Reciprocity (Wage/Effort) Game:
Reciprocity Game Employer Results
Comment: Wages and efforts start high, but a fall in efforts seems to
induce a decline in wages. After 5 periods of fixed matchings, effort had
fallen to zero. In the remaining 5 periods of a random-matching rotation,
wages also fall to zero, which is the Nash prediction for a one-shot game
with these parameters.
- Rent Seeking (Lobbying) Game with three treatments:
Rent Seeking (high lobbying cost)
Rent Seeking (low lobbying cost)
Rent Seeking (high number of competitors)
Comment: The reduction in lobbying cost from $3,000 to $1,000 results in
an approximate doubling of lobbying effort. An increase in the number of
competitors from 2 to 5 reduces lobbying effort per person,
but the amount of rent dissipation rises (as compared with the first treatment that has
the same lobbying cost). The lobbying cost effect is also apparent in the
graph from a different class:
Graph of Average Lobbying Expenditures
- Sequential Search:
Search Results (Individual)
Graph of Accepted and Rejected Draws
Comment: With a search cost of 5, the optimal reservation price for a risk neutral person is 60 cents, which falls
to 30 cents when the search cost is raised to 20 cents in the second treatment.
These predictions are roughly consistent with
the data shown on the results page (for one class) and on the graph (for a different class),
where the orange line tracks an error-minimizing cut-pont that separates accepted and rejected
offer. Risk aversion is likely to result in reservation values that are below
the predictions for risk neutrality, although this effect is not always present, especially
for experiment done in class with hypothetical payments.
- Signaling:
Signaling Results (Individual)
Comment: The experiment was run using the non-neutral "beer-quiche" terminology
to facilitate class discussion. In early rounds, the quiche-eaters receive the
"fight" response and beer-drinkers receive the "flee" response. Behavior for
both "strong" and "weak" types converges to the pooling equilibrium (beer/flee) outcome.
- Statistical Discrimination:
Statistical Discrimination Results (Employers)
Comment: Workers are assigned colors, Green or Purple. Workers see a randomly determined
cost of investing in human capital and then decide whether or not to invest.
The employer can see a worker's color, but not whether a worker invested. The
employer gives a test that provides an imperfect signal about the investment (blue
outcomes are good and red outcomes are bad). Differences in the investment cost
distribution cause green workers to invest more often and get hired more
in the first part of the experiment, and this asymmetry continues in the second
part even after cost distributions have been equalized, as illustrated in the
admin results page shown above. This illustrates
how experienced-based or "statistical" discrimination can arise and continue even
in the absence of underlying biases. This kind of persistent separation is
observed in some sessions and not in others. For a graphs of comparable results in
a different classroom experiment run at Colgate (Fall 2003), see the graphs:
Graph of Employer Hire Rates, by Worker Color
Graph of Worker Investment Rates, by Worker Color
Similar results obtained in a different class of 22 people:
Graph of Employer Hire Rates, by Worker Color
Graph of Worker Investment Rates, by Worker Color
- Traveler's Dilemma:
Traveler's Dilemma Results
Comment: The payoffs for each person in this game are the minimum of the two players' claims,
plus a reward for the low claimant and minus a penalty for the high claimant.
The unique Nash equilibrium is the lowest possible claim, which was 80 in both parts.
With a penalty/reward parameter of 10 in Part I, the claims start high and rise towards
a level of about 170. With a penalty/reward parameter of 40 in Part II, claims
fall towards the Nash prediction. The average claim, shown on the left side of the
display, is 87 in the final period. This experiment was run outside in the grass at Washington
and Lee, using hand-held computers (Compaq IPAQs) with wireless cards.
For results of two sessions from a research experiment reported in
Capra, et al. (1999) "Anomalous Behavior in a
Traveler's Dilemma?":
Graph for Session 1
Graph for Session 2
- Trust Game:
Graph of Data Averages
Comment: One person in each pair receives $10, and whatever is passed
to the other is tripled before the second person decides what to keep and
what to pass back. The graph shows the average amounts passed and returned,
under both random and fixed matchings. The average pass rate is $4-$6 in both
treatments. Do you think putting fixed matchings first would make a difference?
- Vertical Monopoly:
Graph of Average Wholesale and Retail Prices
Comment: In treatment 1, participants are paired into wholesale/retail arrangements, where
the wholesaler's demand is a shift of the retail market marginal revenue curve.
Prices converge to the predicted vertical monopoly levels. All participants have vertically
integrated wholesale/retail markets in treatment 2, and retail prices decline to
the integrated monopoly prediction. In the third treatment, a wholesaler can
try to recover profits via a take-it-or-leave-it franchise fee offer to the retailer.
- Volunteer's Dilemma:
Volunteer's Dilemma (Gains/Losses)
Volunteer's Dilemma (Numbers Effect)
Comment: Each of these classroom experiments has two parts.
The first experiment involved a "neutral" shift in payoffs for the
volunteer and no-volunteer outcomes, with both being gains in one case and with
a possible loss in another. The motivation was to determine whether loss
aversion might increase volunteer rates in the second part. This payoff
shift does not alter the mixed-strategy Nash prediction
that the probability of volunteering is 0.42, which is roughly consistent
with the data in both cases. The second experiment shows that volunteer rates
are lower with larger groups. The identities of volunteers were publicly announced
in the second part of the second experiment, a treatment that had no effect
on the data. A different session with a change in group size from 4 to 6 is
shown in the graph that follows:
Graph of Volunteer's Dilemma for Two Different Group Sizes
- Voting:
Voting Game
Comment: The payoffs generate a "voting cycle." The first treatment implements
a two-stage agenda with options A and B matched in the initial vote, and the
winner then goes against the status quo C. This was a simulation done to test
the software, and the simulated voters voted "sincerely" (non-strategically), which
caused the status quo outcome to win in the final round. The second
treatment implemented an "approval voting" rule that lets each person vote
"approve" or "not approve" for each option. It is possible to approve of more than
one option, which two of the voters did in this simulation.
- Irrigation Reduction Auction:
Irrigation Reduction Results (by individual)
Irrigation Reduction Results (by rank order)
Comment: Opportunity costs are shown in black on the individual results page;
with provisionally accepted bids in green. The bid/cost ratio falls in successive
rounds. The second page shows the
ranked bids by round; notice that the cutoff bid remains about 5% above the
competitive prediction (108) in the final round.
Vecon Lab - June 6, 2025 |