FEELE Laboratory
 

List of Teaching Experiments

Feele homepage

Back to Lecturers: Run Teaching Experiments

Links to Experiment Website


Brief Descriptions of each Experiment

Back to top

  • American Call Option
    The owner of an American call option watches the stock price rise as the expiry date approaches and decides when to exercise or sell the option. Repeat play gives students an understanding of the fair market price for the option and shows that there is no benefit in exercising early.

  • Bertrand Competition
    Sellers compete on price and learn that Bertrand competition drives prices down to marginal cost unless there are only a handful of sellers and opportunities for collusion. The reverse is true when a monopoly is split in two and the consumer ends up paying higher prices.

  • Currency Attack
    Investors attack currencies and depending upon the parameters reach different equilibria: everyone attacks if it is profitable for only a few investors to attack, whereas no one attacks if many investors are required for a profitable attack.

  • Diamond Dyvbig Experiment
    Small investors panic and lose money in a bank run because the bank has insufficient liquidity to pay them all at the same time.

  • Hold-Up Problem
    A buyer is unwilling to make an investment that would be useful only in conjunction with a single supplier because of a fear that the supplier will demand a share of the profits from the investment.

  • Insurance Market with Asymmetric Information
    A consumer and an insurance company both have positive utility from insuring the consumer against a 50% risk of a bad outcome, provided neither of them knows the outcome in advance. If the consumer does know, the asymmetric information leads to the collapse of the market because only the consumers with the bad outcome will buy (adverse selection).

  • Kiyotaki Wright Hazlett Experiment
    Traders in a barter economy suffer a double coincidence of wants problem, which is solved when the good with the lowest storage cost emerges as a common medium of exchange, i.e. money.

  • Lemon Game
    An optimistic buyer, who values an object more than its owner but knows less about its quality, finds out through repeat play that he or she is in a market for lemons and would be better off not to buy at all.

  • Monty Hall Paradox
    Paradox in probability where the disclosure of some apparently irrelevant information causes the relative likelihood of two events to change from 1:1 to 2:1.

  • Network Externalities
    Consumers with different private valuations enter a market with positive externality, where the value of the commodity to each is increasing in the number of others who buy it; the predicted equilibrium is reached where consumers buy when their private valuations exceed a common threshold.

  • Price Discrimination
    A supplier learns by repeated play how to extract the most profit from two buyers with known valuations when different forms of price discrimination are allowed.

  • Team Draft
    Team captains in a sports draft who naively pick their most preferred players end up with a weaker team than those who anticipate the choices of others.

  • Warren Buffett
    Stock market investors find out that a single fund with a high mean return is a risky investment if it also has a high variance; paradoxically, a diversified portfolio containing this fund is a good investment.


American Call Option

Back to top

You may find the example subject instructions helpful.

Abstract

Students play multiple rounds individually against the computer. The student is an investor in the stock market who has bought an American call option on a stock. The stock price starts at £2.00 and increases in 10 random increments of either 2.5% or 7.5%; the call option has a strike price of £2.50. The student watches as the increments are applied and can at any point decide to exercise the option or sell it on the open market. Any profits made in this way increase at a risk-free interest rate of 5% as each remaining increment is applied.

There are two versions of the game. In the first, the student is told the fair market price for selling the option as each increment is applied to the stock price. In the second, the student only learns what the price is after they have made the decision to sell but they are invited to guess beforehand what a fair price might be.

Intended Learning Outcomes

  1. Exercising early does worse than waiting; not worth paying a premium over and above a European call option that can only be exercised on its expiry date.

  2. Discussion of how to calculate the fair market price for selling the option.

Discussion of Likely Results


Bertrand Competition

Back to top

You can quickly log in as a subject to try out this group participation experiment, by pretending to be one of the original participants in a real session. You may also find the example subject instructions helpful.

Abstract

In the standard Bertrand competition game, students play together in small groups as sellers who compete to supply a single buyer (the computer) who always purchases from the seller demanding the lowest price. Sellers choose their prices simultaneously without knowing the prices chosen by others. The goods are perfect substitutes. The demand function is linear, inversely related to price and known to all sellers. The cost of production is the same for all sellers. Students play multiple rounds and there are options for fixed or random pairings.

There is also a setting where the goods are perfect complements. Here, the demand function is once again linear but this time inversely related to the sum of the prices charged by all sellers and the buyer is obliged to purchase an equal quantity from each seller. There is also a monopoly option where each student plays alone against the computer.

Intended Learning Outcomes

  1. Bertrand competition drives prices down to marginal cost in a market with many sellers.

  2. If there are fewer sellers and opportunities for collusion, higher prices may be achievable.

  3. Splitting a monopoly in two can actually be worse for the consumer.

Discussion of Likely Results

In the default setup, sellers have a production cost of £3.00 for each unit sold and may set prices in the range £3.00 to £15.00. The buyer purchases a quantity of 15-P units, where P is the price.

In standard Bertrand competition, P is the lowest price charged. With a small group size (2 or 3 students only) and fixed pairings, there is an opportunity for collusion to develop and P may stay significantly above £3. If this is followed by a second treatment with a larger group size (4 or more students) and random pairings, P quickly drops very close to £3 although the average price may be higher due to failed attempts at collusion.

In the monopoly situation, P is the price set by the monopolist, who makes a profit of (15-P)(P-3), which is maximised by charging a price of £9. Let the monopoly be split in two, where the two duopolists sell perfect complements, each have a production cost of £1.50 and the demand function is 15-(P1+P2), where P1 and P2 are the two prices charged. Now each duopolist i has profit (15-(Pi+Pj))(Pi-1.5) This is maximised at Pi=(16.5-Pj)/2. Thus, we find P1=(16.5-P2)/2 and P2=(16.5-P1)/2. The equilibrium has P1=P2=5.5 so the consumer pays £11 instead of £9. This is suggesting that consumers might not necessarily benefit from anti-trust measures, e.g. breaking up Microsoft into two mini-Bills.


Currency Attack

Back to top

You may find the example subject instructions helpful.

Abstract

Students play together in groups of seven investors who have to decide whether or not to attack a currency. The payoff for attacking a currency is increasing in the number of attackers, whereas the payoffs for not attacking does not depend on the number of attackers.

It pays an individual investor to attack only if sufficient others do so. There is an equilibrium where no one attacks and another one where everyone attacks. The payoffs vary between two settings such that different equilibria are reached for each setting.

Intended Learning Outcomes

  1. Illustrate the relevance of coordination for macroeconomics.

  2. Illustrate multiplicity of equilibria.

  3. Illustrate what determines equilibrium selection.

Discussion of Likely Results

The two settings are with currency A or currency B. The payoffs for not attacking are £5 for currency A and £11 for currency B. The payoff for attacking is £2(N+1), where 0<=N<=6 is the number of others in the group who attack. It is profitable to attack currency A if just 2 others do so, whereas at least 5 others are needed for Currency B. This makes the 'attack' equilibrium more likely for A and the 'no attack' equilibrium more likely for B.

The instructor can configure three different versions of the game, which consists of a number of repeated rounds.

In the 'All Random' game, all the investors attack a single currency, either A or B, selected at random and different each round. Investors are told about the overall number of attacks, as well as the number within their own group of 7, e.g. overall 15 out of 19 attacked A and 4 of 6 others in own group attacked A. This game requires a minimum of 7 students.

In the 'Half Random' game, half of the investors attack currency A and the other half attack currency B, with a random matching of investors and currencies that changes each round. Investors are told about the overall number of attacks on both currencies, as well as the number within their own group of 7, e.g. overall 8 out of 10 attacked A, 1 out of 9 attacked B and 6 of 6 others in own group attacked A. This game requires a minimum of 14 students.

The 'Half Fixed' game is the same as 'Half Random', except that the matching of investors and currencies is fixed, staying the same each round.

Reference

Hazlett has a similar hand-run game.


Diamond Dyvbig Experiment

Back to top

You can quickly log in as a subject to try out this group participation experiment, by pretending to be one of the original participants in a real session. You may also find the example subject instructions helpful.

Abstract

Students play together as a group of small investors who all have money on deposit at the same bank. Each investor has to decide whether to withdraw his/her money today or wait until tomorrow. Some investors are 'impatient' and have a higher utility today, whereas others are 'patient' and have a higher utility tomorrow.

The bank has kept some of the money as cash in hand but the rest is invested in an illiquid asset that does not pay out until tomorrow, with a penalty for early withdrawal today. There is enough money to pay everyone in full provided sufficient of the patient investors decide to wait until tomorrow. Otherwise the money runs out and the bank pays as many investors as it can, with priority being given to those who decided to withdraw today. There is an equilibrium where the patient investors all wait until tomorrow and a second one where they all withdraw today, this latter being a bank run when some investors are not paid because they all try to withdraw early.

Intended Learning Outcomes

  1. Explanation of bank runs (multiple equilibria).

  2. Possible preventative measures.

Discussion of Likely Results

In the default setup, 5 impatient investors and 5 patient investors each invest £1, total £10. Investors choose between withdrawing £1 today or £2 tomorrow. Payoffs represent the investors' utility. Receiving £1 today is worth £1 to both impatient and patient investors. Receiving £2 tomorrow is worth £0.50 to impatient investors and £2 to patient investors. The bank has £5 in cash and £5 invested, with a return on each £1 invested of £0.50 today or £2 tomorrow. There is still some money left to pay investors tomorrow provided no more than two patient investors decide to withdraw today.

Withdrawals Today Investors Paid
Impatient Patient Today Tomorrow
5 0 5 5
5 1 6 3
5 2 7 1
5 3 *7.5 0
5 4 *7.5 0
5 5 *7.5 0
(*) One investor is not paid in full.

A bank run is almost inevitable if the bank gets only £0.20 today and £1.10 tomorrow for each £1 invested. Now the money runs out altogether if only one patient investor panics and decides to withdraw today.

Withdrawals Today Investors Paid
Impatient Patient Today Tomorrow
5 0 5 5
5 1 6 0
5 2 6 0
5 3 6 0
5 4 6 0
5 5 6 0

The bank can prevent a run by suspending payments for today once 5 investors have been paid. If more than 5 investors decide to withdraw today, the bank picks 5 of them at random to be paid; all others who wanted their money today are made to come back tomorrow instead, when they are paid alongside those who preferred to wait originally. So it is possible that some impatient investors will be paid tomorrow if one or more patient investors decide to withdraw today.

Discussion of Application

There was a 'bank run' on hedge funds due to the problems with subprime mortgages. See Paul Krugman: It's A Miserable Life.


Hold-Up Problem

Back to top

You may find the following experiment materials helpful.

Abstract

Students play together in pairs as a buyer and a supplier who are engaged in an ongoing business relationship. The buyer has an opportunity to invest in some specialist equipment that will increase his/her profits but only if the relationship continues with the same supplier. If the investment is made, the supplier, in turn, has an opportunity to take a share of the increased profits by raising his/her prices. If the supplier raises prices, the buyer can either accept the situation or change suppliers but the latter action damages both parties: the buyer has made a wasted investment and the supplier loses the buyer's business.

The hold-up problem arises when the buyer is reluctant to make the investment because of a fear that the supplier will exploit the extra bargaining power.

Intended Learning Outcomes

  1. Origin of the hold-up problem.

  2. Vertical integration as a solution to the hold-up problem.

Discussion of Likely Results

If the investment is not made, the utility payoffs are 0 to both buyer and supplier. If the buyer makes an investment of V and gains increased profits of P, the payoffs are P-V to the buyer and 0 to the supplier, provided the supplier does not raise prices. If the supplier raises prices by R, the payoffs are now P-V-R to the buyer and R to the supplier, provided the buyer does not change suppliers. If the buyer changes suppliers, the payoffs are -V to the buyer and -B to the supplier, where B is an amount reflecting the loss of business.

Students play two consecutive games which differ only in the cost to the buyer of the investment. In both games, P is £1500, R is £750 and B is £1000. In the first game V is £500 whereas in the second game it is £1000. In the first game, if the buyer makes the investment and allows the supplier to raise prices, both parties benefit, with payoffs of £250 to the buyer and £750 to the supplier. In the second game in the same scenario, the payoff to the buyer is £-250, so the buyer should not make the investment.


Insurance Market with Asymmetric Information

Back to top

You may find the example subject instructions helpful.

Abstract

Students play together in pairs as a consumer and an insurance company. The consumer has to decide whether or not to buy insurance and the company has to decide whether or not to sell insurance. The two decisions are made simultaneously. There is a 50% risk of a 'bad' outcome (from the consumer's point of view), when the consumer would benefit from buying insurance. The payoff matrices for the good and bad outcomes are as follows.

GOOD Company Sells Insurance
Yes No
Consumer Buys Insurance Yes 0.6, 0.6 1, 0
No 1, 0 1, 0

BAD Company Sells Insurance
Yes No
Consumer Buys Insurance Yes 0.6, -0.4 0, 0
No 0, 0 0, 0

The bad outcome costs the consumer 1 if he/she does not have insurance. The good or bad outcomes only affect the insurance company if it sells insurance and the consumer buys it. The insurance costs 0.6 to fully cover the loss of 1, so the payoff to the company is 0.6 for the good outcome and -0.4 for the bad outcome. The consumer is risk averse and has extra utility of 0.2 for the security of being insured, so the consumer's payoff when insured is 0.6 for both the good and bad outcomes.

The instructor can configure two versions of the game, one where neither player knows whether the outcome is good or bad and another where only one player is informed - in this case the consumer. Neutral terminology is used whereby the consumer and insurance company are referred to as the row player and column player respectively.

Intended Learning Outcomes

  1. Asymmetric information leads to adverse selection in an insurance market.

  2. It is not always good for the consumer to be better informed.

  3. Characteristics of insurance markets in general.

Discussion of Likely Results

If neither player knows whether the outcome is good or bad (and the two outcomes are equally likely), the consumer is better off buying insurance and the company is better off selling it. The consumer has an expected payoff of 0.6 when insured and 0.5 when uninsured. The company has an expected payoff of 0.1 when the consumer is insured and 0 when the consumer is uninsured. So there is profit in the market for both players

If, however, the consumer knows whether the outcome is good or bad, the company is better off not selling insurance. The consumer is better off only buying insurance when the outcome is bad, so the company can only sell to those consumers who are a bad risk. The company now has an expected payoff of -0.4 when the consumer is insured and 0 when the consumer is uninsured, so it cannot make a profit and the market for insurance collapses. Since insurance also benefits the consumer, this outcome is bad for both players.

For insurance to be profitable, the company needs to balance the risk across a large random sample of consumers. Adverse selection disrupts this by leaving the company with a disproportionately large number of consumers who are a bad risk, similar to a lemon market where the bad business drives out the good. (Alongside this is moral hazard, the tendency of consumers who are insured to act more recklessly.) To counter this, insurance companies require consumers to disclose factors that make them a bad risk, e.g. smokers have to pay more for health insurance. Insurance may then become prohibitively expensive for certain groups of consumers.


Kiyotaki Wright Hazlett Experiment

Back to top

You may find the example subject instructions helpful.

Abstract

Students play together in a single large group as agents who meet randomly in pairs over the course of a number of rounds. There are 3 types of agents and 3 goods and each agent produces a different good from what they consume, so they must trade in order to consume. The experiment is designed so that trading can only occur if at least one agent in each pair is willing to accept a good that is not his/her own consumption good. Consequently the good with the lowest storage cost spontaneously emerges as the generally accepted medium of exchange.

Intended Learning Outcomes

  1. Origin of money.

Discussion of Likely Results

There are roughly equal numbers of Type 1, 2 and 3 agents and each starts round 1 with 40 points and one unit of the good that they produce.

Agent Consumes Consumption Bonus Produces Storage Cost Net Payoff
Type 1 Good 1 20 points Good 2 4 points 16 points
Type 2 Good 2 20 points Good 3 9 points 11 points
Type 3 Good 3 20 points Good 1 1 point 19 points

Agents earn the net payoff shown above when they trade for their own consumption good, otherwise they must pay the storage cost of whatever good they hold, whether they have traded in that round or not.

Good 1 is the least expensive to store and therefore emerges as 'money'.

Type 2 agents are at a disadvantage because they produce Good 3 which is the most costly to store and will usually only be accepted by a Type 3 agent. If Good 3 were less expensive to store, some Type 1 agents might accept it speculatively in the hope of quickly offloading it to a Type 3 agent holding Good 1.

Further Discussion Points

  1. The inefficiency of a barter economy.

  2. Money helps solve the problem of double coincidence of wants.

  3. Fun discussion: some people may do quite well in a barter economy, see Kyle MacDonald: One Red Paperclip.


Lemon Game

Back to top

You can quickly log in as a subject to try out this individual-progress experiment. You may also find the example subject instructions helpful.

Abstract

Students play individually as a buyer who bids to purchase an object from the computer in the role of seller. The seller knows the value of the object and will not sell it if the buyer bids less than it is worth. The buyer does not know the value of the object, knowing only the interval over which the value is uniformly distributed. The buyer values the object more than the seller and is willing to pay up to 1.5 times its value. Should the buyer bid and if so how much?

Intended Learning Outcomes

  1. Concept of a lemon market and criteria for one to exist.

  2. Preventative measures: lemon laws.

Discussion of Likely Results

In the default setup, values are uniformly distributed over the interval £0 to £1. If the buyer makes a bid of b, the seller will only sell the object if the value to the seller is less than b. Thus, if it is sold it will on average have a value of b/2. The object is worth 1.5(b/2) = 3b/4 to the buyer, so the buyer's profit is -b/4 and the conclusion is that the buyer should not bid at all (bid 0).

The market for second hand cars is an example of a lemon market. There is asymmetry of information because the seller knows more about the quality of the car than the buyer. The seller of a bad car has a strong incentive to sell it at a much higher price than it is worth and the buyer is insufficiently protected by regulation or warranties in this case. Conversely, it is difficult for the seller of a good car to demonstrate its quality to the buyer. The result is that bad cars tend to out-number good ones in the market. This is an explanation as to why the price of a new car drops so rapidly once it leaves the showroom; see The Market for 'Lemons': Quality Uncertainty and the Market Mechanism by Akerlof (1970).


Monty Hall Paradox

Back to top

You can quickly log in as a subject to try out this individual-progress experiment. You may also find the example subject instructions helpful.

Abstract

Students play multiple rounds individually against the computer and in each round try to locate a prize that has been hidden at random in one of three closed boxes. The student starts by guessing where the prize is, after which the computer opens one of the other two boxes that is empty and gives the student the option of sticking with his/her original choice or changing to the remaining unopened box. This game is the well-known Monty Hall game show paradox where the student is the contestant and the computer is Monty.

Changing is twice as likely to be successful as sticking. Because this is so counter-intuitive, the instructor may also configure a repeat-play 'strategy' version of the game, where the student plays the game once in each round as before but then the computer plays the game a further large number of times using the student's choice of initial box and 'strategy' of 'stick' or 'change'. The computer displays the results of the individual games plus a summary.

There are also versions of the game with four and five boxes.

Intended Learning Outcomes

  1. Gain an improved understanding of probability and Bayes' rule.

  2. Understand the origins of the paradox.

Discussion of How the Paradox Arises in the 3 Box Game

The original guess plainly has a 1/3 probability of being correct. The paradox arises principally because it does not seem as if the computer is imparting any useful information by opening an empty box. However there is a 2/3 probability of the prize being in one of the two boxes that were not guessed, so after the computer helpfully eliminates one of them, the 2/3 probability remains attached to the remaining unopened box, which is therefore twice as likely to contain the prize.


Network Externalities

Back to top

You can quickly log in as a subject to try out this group participation experiment, by pretending to be one of the original participants in a real session. You may also find the example subject instructions helpful.

Abstract

Students play together in a single large group as consumers who must simultaneously decide whether or not to buy a commodity. The price of the commodity is the same for all and is common knowledge. The value of the commodity to each consumer is equal to that consumer's private value multiplied by the proportion of other consumers who decide to buy. Private values are uniformly distributed across a known interval.

This is an example of a positive network externality, where the entry of additional consumers into a market has a beneficial effect on those who already possess the commodity, e.g. the market for fax machines, which are only useful if there are enough other people you can send a fax to.

Intended Learning Outcomes

  1. Network externalities.

Discussion of Likely Results

One should only buy the good if the expected value is above the price E[n*V]>=p (where n is the proportion of others buying the good). There are multiple equilibria. One equilibrium has no one buying the good. For instance, if no one buys a fax a machine, then you shouldn't buy a fax machine E[0*V]<p.

In the default setup, the price is £2.40 and the private values are uniformly distributed in the interval [£0.00, £10.00]. Here there is another equilibrium with a positive n. Those deciding to buy the good have should have V such that E[n]V>p. In words, they each decide to buy when for private values above a common threshold value V=p/E[n]. What should that value be? The proportion of others who buy is approximately (10-V)/10, so each consumer is indifferent between buying and not buying when V(10-V)/10=2.4, which has solutions for V=4 and V=6. When V=4, this is the success equilibrium where 60% of the consumers buy the good. When V=6, it is in an unstable equilibria, which one can call the "tipping point". If people with values of 5.9 start buying it then, the object would be a hit, that is, it will go to the equilibrium with V=4. This is because the proportion of those buying it is now (10-5.9)/10=0.41 and p/0.41=2.4/0.41=5.85. So those with values above 5.85 will start buying the good. Repeating this calculation (an infinite number of times) one will arrive at V=4.

Further Reading

Bergstrom and Miller, Experimental Handbook.

Varian, Hal, Intermediate Microeconomics

Malcolm Gladwell, 'The Tipping Point'


Price Discrimination

Back to top

You can quickly log in as a subject to try out this individual-progress experiment. You may also find the example subject instructions helpful.

Abstract

Students play individually in the role of a supplier, whose objective is to extract as much profit as possible from two buyers played by the computer, when various forms of price discrimination are allowed. The buyers' valuations are different and the supplier knows the value to each buyer of 1 item and of 2 items, i.e. there are four valuations in all. Students play four consecutive games where different degrees of price discrimination are allowed: no discrimination, partial discrimination based on buyer (3rd degree), partial discrimination on quantity (2nd degree) and full discrimination on both buyer and quantity (1st degree). Playing multiple rounds of each game allows students to experiment with different prices and understand how the buyers behave.

Intended Learning Outcomes

  1. Understand price discrimination and how to calculate buyer surpluses.

  2. Price discrimination is good for the supplier and the fully discriminating monopolist makes the most profit.

  3. Suppliers may have perverse incentives that are unhelpful to some consumers, e.g. an airline deciding not to improve the uncomfortable seating in economy class to encourage business customers to pay more to get a premium service. This is profitable if business customers are more sensitive to comfort and tourists to price.

Discussion of Likely Results

Each item costs £5 to produce and the buyers' valuations are as follows.

Buyer 1 item 2 items
A £20 £20
B £30 £40

Buyers prefer to buy 2 items rather than 1 if indifferent in terms of their valuations.

With no discrimination, charge £20 per item and sell 1 to A and 1 to B, profit £30.

With buyer discrimination, charge A £20 per item and B £30 per item and sell 1 to A and 1 to B, profit £40.

With quantity discrimination, charge £20 for 1 item and £30 for 2 items and sell 1 to A and 2 to B, profit £35. (B has the same surplus of £10 for both 1 item and 2 items and therefore buys the larger quantity.) It is easy for the students to think that the prices should be £20 for 1 item and £40 for 2 items. The key for them to understand this is that then buyer B will buyer only 1 item and get a surplus of £10 rather than a surplus of 0 for buying 2 items.

With full discrimination, charge A £20 for 1 item and B £40 for 2 items and sell 1 to A and 2 to B, profit £45. 3. One could mention that while full discrimination is worst for the consumer, it is the most efficient (highest buyer+seller surplus).


Team Draft

Back to top

You can quickly log in as a subject to try out this group participation experiment, by playing alone against the computer, which always makes its preferred choice. You may also find the example subject instructions helpful.

Abstract

Students play together in pairs or threes and take turns choosing objects from a pool. Once each object has been chosen it is removed from the pool and cannot be chosen a second time by any player. The analogy is with a sports draft with the players as team selectors and the objects as footballers. The players have a valuation for each object with a strict preference ordering which is different for each player. These valuations are common knowledge.

There are versions of the game for 2 players choosing 2 objects each (total 4 objects), 2 players choosing 3 objects each (total 6 objects) and 3 players choosing 2 objects each (total 6 objects). There is also a 'solo player' option where students play individually against the computer, which takes the role of the other player(s) and uses the sincere strategy of always choosing the object with the highest valuation.

Intended Learning Outcomes

  1. Subgame perfection, backward induction.

  2. More advanced students can learn the strange result of Brams and Straffin where such a selection can make everyone worse off.

Discussion of Likely Results

The default setup is for 2 players choosing 2 objects each and appears in the paper by Brams and Straffin.

Player 1 Player 2
A:£4.00 B:£4.00
B:£3.00 C:£3.00
C:£2.00 D:£2.00
D:£1.00 A:£1.00

Consider first what happens if player 1 plays sincerely and chooses object A first. Note that player 2 cannot now do any better than to choose object B first, after which player 1 will pick C, leaving player 2 with B and D for a payoff of £6 and player 1 with A and C for a payoff of £6. (If instead player 2 were to choose either C or D first, player 1 would pick B, leaving player 2 with C and D for a smaller payoff of £5.)

Now consider what happens if player 1 chooses object B first. Now player 2's most valuable object has gone and he cannot do any better than to end up with both C and D for a payoff of £5. So it does not profit player 2 to pick object A and he should therefore pick either C or D first. So now player 1 ends up with A and B for a payoff of £7 and player 2 ends up with C and D for a payoff of £5.

Further Points

Such a selection has been used as a method for conflict resolution between political parties. Was used as a part of the Good Friday agreements.

Further Reading

Brams and Straffin. Prisoners' Dilemma and Professional Sports Drafts.

Brams and Kaplan. 'Dividing the Indivisible: Procedures for Allocating Cabinet Ministries to Political Parties in a Parliamentary System,' Journal of Theoretical Politics, 2004 16 (2):143-173.

Brams. 'Mathematics and Democracy' 2007, Princeton University Press.

O'Leary, Brendan, Bernard Grofman and Jorgen Elklit (2001). 'The Use of Divisor Mechanisms to Allocate and Sequence Ministerial Portfolio Allocations: Theory and Evidence from Northern Ireland.' Mimeo, Department of Political Science, University of Pennsylvania.


Warren Buffett

Back to top

You can quickly log in as a subject to try out this individual-progress experiment. You may also find the example subject instructions helpful.

Abstract

Students play individually and have a choice of 3 funds in which to invest money. The first fund mimics the long term behaviour of the stock market and exhibits steady growth with occasional downturns. The second fund has by far the highest mean gross return, trebling in value half of the time, but it also has the highest variance and is a risky investment. The third fund mimics the inflation-adjusted behaviour of treasury bonds and stays virtually constant.

The game consists of a number of repeated rounds and the growth in each fund during a given round is determined by making a random selection from among 6 possible outcomes. In the basic game, students must invest 100% in one of the 3 funds, although they may switch funds between rounds. Alternatively, the instructor may allow investments to be split across more than one fund.

Intended Learning Outcomes

  1. Variance as a measure of the risk of an investment; volatility drag.

  2. Role of a diversified portfolio in reducing risk.

Discussion of Likely Results

The Green 'stock market' fund is the best of the original investments, when adjusted for volatility.

The Red fund is superficially attractive because of its high mean return and one or two students who invest in it over 20 rounds can be expected to do spectacularly well. However the majority will be ruined and overall it represents a poor investment.

The Blue 'treasury bond' fund is a safe investment but with very uninspiring growth.

Outcome Green Red Blue Purple
1 0.8 0.05 0.95
2 0.9 0.2 1
3 1.1 1 1
4 1.1 3 1
5 1.2 3 1
6 1.4 3 1.1
Mean return 1.0833 1.7083 1.0083 1.3583
Variance of return 0.0381 1.7554 0.0020 0.4393
Volatility adjusted 1.0643 0.8307 1.0073 1.1387

Purple is a portfolio investment of 50% in the risky Red fund and 50% in the safe Blue fund. Critically, the portfolio is re-balanced between rounds to retain 50% of the total value in each fund. Since Red and Blue are independent random variables, and the variance of Blue is negligible, the variance of Purple is pretty much one quarter that of Red.

mean(Purple) = mean((Red + Blue) / 2) = (mean(Red) + mean(Blue)) / 2

var(Purple) = var((Red + Blue) / 2) = (var(Red) + var(Blue)) / 4

So, paradoxically, Purple is a better investment than Green, despite being a mixture of two funds that are worse performers individually.

Acknowledgements

Being Warren Buffett: A Classroom Simulation of Risk and Wealth when Investing in the Stock Market