a natural experiment in the prisoner's dilemma

* This working paper represents a work in progress, circulated to encourage discussion and comments, and should be read as

such. This work should not be quoted without permission from the author. Any opinions expressed in this work are those of

the author and do not necessarily reflect the views of the Department of Economics, University College Cork.

DEPARTMENT OF ECONOMICS

UNIVERSITY COLLEGE CORK

WORKING PAPER SERIES

SPLIT OR STEAL? A NATURAL EXPERIMENT OF THE

PRISONER’S DILEMMA.

Working Paper: 09-XX*

Seamus Coffey

Health Economics Group

Department of Economics

University College Cork

ABSTRACT: This paper uses the final round of the UK TV game Goldenballs as a natural

experiment to analyse the choices made by people when faced with a prisoner‟s dilemma type

situation. In the game two contestants make a „split‟ or „steal‟ to decide how a jackpot of

varying size is to be distributed – split, stolen or lost. Players cooperate 48% of the time with

males cooperating more than females and young players cooperating more than mature

players. There is considerably more cooperation in games between genders than in games

with players of the same gender. Players in the same age category cooperate more with each

other than players in different age categories. Mature players are the most efficient players at

converting jackpots into winnings.

JEL Classification Numbers: C72, C93, D64

Keywords: Prisoner‟s Dilemma, Natural Experiment, Cooperation, Gender Differences, Age

Differences.

Correspondence:

Address: Department of Economics, University College Cork, Cork City, Ireland.

Email: [email protected]

Telephone: 353 21 4901928

Fax: 353 21 4273920

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1. Introduction

The TV game show Goldenballs concludes with two contestants facing off in a situation that

is a variation of classic set-up of The Prisoner‟s Dilemma. The Prisoner‟s Dilemma is the

most frequently used example in analysing situations where people will benefit from co-

operating but have an individual incentive for non-cooperation. Using data from the show

this paper considers the characteristics of people who choose to cooperate, and the impact, if

any, that the characteristics of their opponent have.

Overall, players cooperate 48% of the time with males cooperating more than females and

young players cooperating more than mature players. There is considerably more

cooperation in games between genders than in games with players of the same gender.

Players in the same age category cooperate more with each other than players in different age

categories. Mature players are the most efficient players at converting jackpots into

winnings.

2. The Prisoner’s Dilemma

Following Ryan and Coffey (2006) the game is generally described using the following

analogy:

Two prisoner‟s have been arrested under the suspicion of having committed murder and are

placed in separate isolation cells. Both care much more about their personal freedom than

about the welfare of their accomplice. The police have insufficient evidence for a conviction

and offer each of the prisoners the same deal: 1

„You may choose to confess or remain silent. If you confess and your accomplice

remains silent I will drop all charges against you and use your testimony to ensure

that your accomplice receives the full 25-year sentence. Likewise, if your accomplice

confesses while you remain silent, they will go free while you do the time. If you both

confess I get two convictions, but I'll see to it that you both get early releases after ten

years. If you both remain silent, I'll have to settle for 4 year sentence on firearms

possession charges.‟

1 This situation is set up and described as a Prisoner‟s Dilemma in the 2002 film Murder by Numbers when two

suspects are arrested and questioned on suspicion of murder in the manner described here.

3

Each prisoner must make the choice of whether to remain silent (co-operate with his

accomplice) or confess (defect and betray his accomplice). A one-shot, two-player prisoners‟

dilemma can be summarized as follows:

Table 1: Payoff matrix for the classic prisoner‟s dilemma

Prisoner 2

Confess Stay Silent

Prisoner 1 Confess 10yrs, 10yrs free, 25yrs

Stay Silent 25yrs, free 4yrs, 4yrs

The dilemma arises when one assumes that both prisoners only care about minimising their

own jail terms, i.e. that is they are seeking to minimise the numbers in the above pay-off

matrix. In each cell the first prison sentence listed corresponds to the row player, Prisoner 1,

and the second prison sentence corresponds to the payoff for the column player, Prisoner 2.

We can see that the outcome of each choice for a prisoner depends on the choice of the

accomplice.

The problem with the Prisoners‟ Dilemma is that if both decision-makers were purely

rational, they would never cooperate. If Prisoner 1 assumes that Prisoner 2 will confess he

should also confess, giving a 10 year sentence rather than the 25 years for remaining silent.

If he assumes that Prisoner 2 will remain silent his best course of action is also to confess as

this will mean no jail time versus four years for remaining silent. Thus, we see that for

Prisoner 1 non-co-operation with his accomplice or confessing is his dominant strategy. A

similar analysis for Prisoner 2 will show that confess of also a dominant strategy for him.

Thus the Nash Equilibrium for this game is for both prisoners to confess and each receives a

jail sentence of ten years. It is easy to see that this is not the best collective outcome for the

prisoners.

If reasoned from the perspective of the optimal outcome the correct choice would be for the

prisoners to cooperate with each other and deny the allegations, as this would reduce the total

jail time served. Any other decision would be worse for the two prisoners considered

together. When the prisoners both confess, each prisoner achieves a worse outcome than if

4

they had both denied. This demonstrates that in a non-zero sum game the Pareto optimum and

the Nash equilibrium can be opposite.

The only way to achieve the Pareto optimal solution in the one-shot Prisoners‟ Dilemma is if

a prior agreement to deny is somehow enforceable. This would clearly be in the prisoner‟s

joint interests. Unless the agreement to deny is enforced in some way the incentive for both

prisoners to confess is so strong that neither can trust the other to keep to any agreement. If

Prisoner 1 sticks to the agreement, Prisoner 2 can go free by defecting on the agreement and

confessing.

A significant amount of research on the Prisoners‟ Dilemma relates to evidence of collusion

and cooperative behaviour. This type of behaviour contradicts the theoretical prediction that

non-co-operation is the dominant strategy. For example, large firms can and do collude. In

an experimental setting Camerer (2003) points out that people playing one-shot Prisoners‟

Dilemma games cooperate around fifty percent of the time.

3. The Goldenballs Dilemma

Several researchers have used television game shows provide a natural venue to observe real

decisions in an environment with high stakes. For example, in the U.S., Berk, Hughson, and

Vandezande (1996) study contestants‟ behaviour on The Price is Right to investigate rational

decision theory, Gertner (1993) and Beetsma and Schotman (2001) make use of data from

Card Sharks and Lingo, respectively, to examine individual risk preferences and, finally,

Metrick (1995) uses data from Jeopardy! to analyse behaviour under uncertainty and players‟

ability to choose strategic best responses.

The example chosen here is from series one of the UK game show Goldenballs. The dataset

comprises the entire 40 episodes broadcast between June and August 2007. All 40 episodes

were recorded before the show began screening. It is the final element of the game “split or

steal” that is our primary focus but what follows is a brief description of how the final two

players are chosen and the amount of the jackpot they will be playing for.

Round 1: Each show begins with four players, two male and two female and a drum

containing 100 „golden balls‟ with cash values ranging from £10 to £75,000. 12 balls are

drawn at random from this drum and these along with four „killer‟ balls are distributed

5

between the four contestants.2 Each contestant has four balls. The contents (cash value or

„killer‟) of two are visible to all players, while the contents of the remaining two balls are

visible only to their owner.

In turn, the contestants announce the contents of their hidden golden balls. They can either

tell the truth or lie about their amounts. After each contestant has done this, they discuss who

they think is lying and try to establish who has the worst set of golden balls, either in terms of

having the lowest amount of money or the most „killer‟ balls.

The contestants then secretly vote for which of them they would like to leave the game and

the player who receives the most votes is eliminated. At the end of the round, each contestant

reveals the contents of the golden balls on their back row and the eliminated contestant's

golden balls are "binned", and are out of the game for good.

Round 2: The three remaining contestants' golden balls are put back into the drum, along

with two more cash balls, as well as one more „killer‟ ball, leaving fifteen golden balls in

play. These fifteen golden balls are split among the remaining three contestants randomly.

Again the contents of two of the balls are visible to all players with the contents of the

remaining three hidden. The game proceeds are per Round 1 with a secret vote determining

the player to be eliminated.

Bin or Win: The remaining ten balls plus one additional „killer‟ are placed on a table balls.

The players take it turn to select a ball to "bin" (eliminate from the game) and a ball to "win"

(add to the jackpot). Cash values are added to the jackpot. If a „killer‟ ball is picked to be

won, then the accumulative value of the jackpot is divided by 10. This process is repeated

five times.

Split or Steal: It is at this stage that the contestants face a decision similar to the Prisoner‟s

Dilemma as they have to make a decision about the final jackpot. Each contestant chooses a

ball, either „split‟, which means they try and split the jackpot with the other contestant or

„steal‟ which means they try and steal the entire jackpot for themselves. There are three

outcomes as follows:

2 At the end of the game if a „killer‟ ball remains and is chosen as one of the five balls that will make up the

value of the jackpot, the „killer‟ ball will result in the jackpot being ten times smaller.

6

Both players choose „split‟: The winnings are split equally between them.

One player chooses „steal‟, one „split‟: The player who played „steal‟ gets all the

money.

Both players choose „steal‟: No-one gets any money.

The problem is the same as The Prisoner‟s Dilemma except it is not quite as pure. This is a

one shot game, but the players are in the same room, in fact, they‟re looking right at each

other, their friends and family are watching and they are given the opportunity to convince

the other person of their intention to either „split‟ or „steal‟. There is more at stake than some

money, their reputation amongst all people for one. On top of all of this they have been

playing a game for the past half hour and have had the chance to betray each other already.

The similarities with the Prisoner's Dilemma are:

1. It is a game of cooperation (split) or defection (steal).

2. Decisions are made simultaneously.

3. It is a one shot game

The major differences are:

1. This is a zero-sum game.

2. The players can communicate.

3. Steal (defect) is only a weakly dominant strategy

Each player has an incentive to defect and play „steal‟ because he is never worse off

monetarily for doing so. Table 2 is a payoff matrix for the game.

Table 2: Payoff matrix for the Goldenballs „Split‟ or „Steal‟ round

Player 2

Steal Split

Player 1 Steal 0%, 0% 100%, 0%

Split 0%, 100% 50%, 50%

The worst outcome in this game is for the players to both choose „steal‟ as that would mean

no one wins the jackpot. All other scenarios mean the full jackpot is given to at least one of

the players. At initial inspection it may appear that the jackpot will be given out ¾ times and

no jackpot a ¼ of the time. But the interesting thing with this game is that assuming all

7

players behave rationally the outcome will actually always be that no one wins the jackpot

(i.e. two steals).

If you put yourself in the position as a player, you can see how this works. There are two

possible options that your opponent can choose („steal‟ or „split‟).

Take scenario 1 where your opponent chooses „split‟. Here if you choose „split‟ you will get

half the jackpot, if you choose „steal‟ you will get the entire jackpot. So obviously, any

rational person will choose „steal‟ as this will maximise their winnings.

Take scenario 2 where your opponent chooses „steal‟, in this scenario it is irrelevant whether

you choose „steal‟ or „split‟ because either way you will get nothing. So given the scenario 2

decision is irrelevant (as „steal‟ and „split‟ both result in 0) your decision should be based

purely on scenario 1 where it has already been illustrated that any rational person will choose

„steal‟.

So the optimum strategy for any player is „steal‟. Of course the problem with this is that your

opponent has the same options as you and therefore will pick „steal‟ which means the game

ends in two „steals‟. So going back to the game show assuming that all participants are

rational human beings the first 55 minutes of the show are irrelevant because whatever the

jackpot ends up being the result of the game will always end up with no one wining anything.

So what actually happens when people are faced with this choice on the show? The show is

currently half way through its sixth series and, in the five and a half series to date, 253

episodes have been broadcast. The paper uses data on the 40 episodes in series one that were

broadcast in 2007. This gives us a sample of 80 people who were presented with the

Goldenballs Dilemma.

List (2006) provides a number of useful caveats when considering data from a game show

setting. First, those who appear on the show may not be drawn randomly from the population

of interest. Second, the public nature of the play may affect behaviour so that people do not

consider simply a one-off game with the other contestant but as part of a repeated game with

those who view the show.

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4. The Data

Summary statistics of the 80 participants in the sample are provided in table 3. This provides

an overview of the amounts earned in the first three rounds, i.e. the jackpot played for, the

cooperation rates and the average amount of money won. The final column is a measure of

the participants‟ ability to transform jackpots into winnings, the efficiency rate.3

Even though we have shown that 'steal' is the weakly dominant strategy of the 80 contestants,

42 of them chose 'split', or just over 52%, with the other 38 contestants obviously choosing

'steal'. This is in line with previous findings of cooperation rates in other trials and

experiments of the prisoner‟s dilemma.

Table 3: Summary of participants‟ characteristics, choices and outcomes

N % Average

Jackpot

(Std. Dev.)

Cooperation

Rate

Average

Winnings

(Std. Dev.)

Average

Winnings /

½ Average

Jackpot

Overall 80 - £12,976

(15,992)

0.52 £5,395

(11,511)

0.83

Male 37 46% £9,192

(12,990)

0.46 £4,320

(9,555)

0.94

Female 43 54% £16,231

(17,690)

0.58 £6,320

(13,004)

0.78

White 76 95% £12,944

(16,014)

0.51 £5,333

(11,667)

0.82

Non-White 4 5% £13,587

(17,952)

0.75 £6568

(9181)

0.97

Young 37 46% £11,480

(14,990)

0.49 £2,469

(5,845)

0.43

Mature 43 54% £14,262

(16,874)

0.56 £7,912

(14,350)

1.11

Of the 43 females who made it to the final round, 24 (or 58%) chose „split‟, while of the 37

males only 17 (or 46%) chose „split‟. Female had higher average winnings than males, but

3 This figure will lie between zero and two. A figure of one would mean that on average each member of this

group won half of the available jackpot. A figure of less than one indicates that the average winning was less

than half of the average jackpot meaning that some jackpots were lost or stolen on this group. A figure of

greater than one means that this group won more than half the jackpot on average, meaning there were some

successful stealers in this group and relatively fewer suckers who had jackpots stolen on them. A figure of two

would mean that all members this group successfully stole the jackpots they played for. If there are games

between members of the same group the maximum efficiency figure will be less than two.

9

this is primarily because they played for bigger jackpots. If we look at efficiency rates males

have a rate of 0.94, while for females the figure is only 0.78.

The only group who had an efficiency rate of greater than one, that is their average winnings

were greater than half the average jackpot played for, were the mature group with an

efficiency rate of 1.11. In contrast the young participants had the worst efficiency rate of

only 0.43. On average they won less than a quarter of the total jackpot amounts they played

for.

The average jackpot competed for in the 40 episodes was £12,975.76, ranging from just £3 to

£61,060. Table 4 gives further details on the jackpots and the actual outcomes of the 40

games played.

Table 4: Summary of jackpots played for

Outcome N % Average

Jackpot

Standard

Deviation

Minimum Median Maximum

All Games 40 - £12,976 15,992 £3 £7,108 £61,060

Lost 10 25% £8,742 14,695 £455 £3,815 £50,450

Stolen 18 45% £17,807 18,308 £3 £13,265 £61,060

Split 12 30% £9,245 111,06 £32 £5,109 £38,950

There were 12 episodes in which both contestants chose 'split' and the jackpot was divided

between them. The average split jackpot was £9,245.49. That leaves 18 people choosing

'split' who had 'steal' played against them and ended up with nothing. The average stolen

jackpot was £17,807.14. In the remaining ten episodes both contestants choose 'steal' and the

jackpot was lost. The average lost jackpot was £8,742.25.

Across the 40 games a total prize fund of £519,030.50 was played for. The 10 “lost” jackpots

came to a total of £87,422.50. This means our 80 contestants had an efficiency rate of 0.83.

17% of the total available winnings were lost due to non-cooperation by both participants.

If strategies were played randomly we would expect the jackpot to be split 25% of the time,

stolen 50% of the time and lost 25% of the time. The actual percentages of 30%, 45% and

25% only differ ever slightly from this with slightly more splits than steals as predicted by

purely random behaviour.

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5. Decision Factors

We will now consider a number of factors that may have an impact on the decisions the

players make in the „split‟ or „steal‟ round. The factors considered include, the size of the

jackpot, gender and gender of opponent, age and age of opponent, profession and hair colour.

Size of Jackpot: In games with the 12 biggest jackpots (£61,060 to £16,600, average

£32,968.33) „split‟ is played 13 times. In games with the 12 smallest jackpots (£3 to £1,815,

average £755.58) „split‟ is played is played 12 times. This is 54% and 50% of the time in

each case. This suggests that the size of jackpot is not a significant determinant of the

strategy played. If we look at the outcomes of the 12 biggest jackpots, 9 are successfully

stolen (75%), with 2 split and 1 lost. Of the 12 smallest jackpots only 4 are successfully

stolen (33%) with 4 split and 4 lost.

Gender Differences: Of the 40 games, 23 were male versus female, 7 were male versus

male and 10 were female versus female. These are summarised in Table 5.

Table 5: Outcomes of games by gender

Male Female

Male

Number = 7

Lost = 3; Stolen = 1; Split = 3

Average Jackpot = £3,478

Lost = £2,935; Stolen = £648; Split = £13,600

Cooperation Rate = 0.35

Average Winnings = £1,110

Efficiency Rate = 0.64

Number = 23



Lost = £11,125; Stolen= £17,377; Split = £9,516


Male = 0.52; Female = 0.61


Male = £6,275; Female = £3,494


Male = 0.99; Female = 0.55

Female

Number = 10

Lost = 1; Stolen = 7, Split = 2


Lost = £11,872 ; Stolen = £25,653 ; Split = £5,916




11

Each quadrant represents the different types of game (male versus male, male versus female,

female versus female) as indicated by the row and column markers. The number of each type

of game is given as well as the breakdown of split, stolen or lost outcomes of these games.

The average jackpot played for, the co-operation rate of the participants, the average winning

and the efficiency rate for each type of game is given. Additional data by gender is given for

male versus female games.

Against females, females played „split‟ 55% of the time and played it 61% of the time against

males. Males played „split‟ 52% of the time against females but only 35% of the time against

males. There is noticeably more cooperation across genders than amongst genders.

Of the 12 games where the jackpot was split, 9 were in games where there was a male and a

female (40% of male versus female games), while only 1 was in an all male game (14% of all

male games) and 2 were in all female games (20% of all female games). In the 17 games of

the same gender the jackpot was split only 3 times (18% of same gender games).

70% of female versus female games resulted in a successful „steal‟! With only 10% of

jackpots lost, female versus female games were the most efficient, though clearly not the

most equitable. The amount lost was only 6% in all female games, but this is largely due to

the high rate of successful steals. 43% of male versus male games ended in a successful

steal, but with 43% of jackpots also lost the efficiency rate of male versus male games was

only 0.64. Of the 8 steals in the male versus female games (34% of such games), 5 were by

males and 3 by females. The overall efficiency rate in male versus female games was 0.77,

but males fared substantially better with a rate of 0.99 against 0.55 for females.

Age Differences: The players were broken into two age categories. “Young” are those

players who are less than 30. “Mature” are players above 30. 37 players are the young

category with 43 in the mature category. There were 11 games between two young

contestants, 14 games between two mature contestants and 15 of the games featured a young

player against a mature player. The breakdown of these games by age category is in table 6.

Against young opponents, young players played „split‟ 55% of the time and played it 40% of

the time against mature opponents. Mature players played „split‟ 75% of the time against

other mature players but only 20% of the time against young players. There is noticeably

12

more cooperation amongst the age categories than between them, particularly in the mature

age category.

Table 6: Outcome of games by age

Young Mature

Young

Number = 11



Lost = £17,320; Stolen = £7,105; Split = £5,551




Number = 15



Lost = £3,024; Stolen = £20,462; Split = n/a


Young = 0.40; Mature = 0.20


Young = £3,293; Mature = £8,984


Young = 0.48; Mature = 1.33

Mature

Number = 14



Lost = n/a; Stolen = £17,451; Split = £11,904




None of the 15 games between a young player and a mature player resulted in a split pot.

Young players split 5 of their 11 games (45%) and mature players split 7 of their 14 games

(50%).

The efficiency rate of young players is very low. In games amongst themselves young

players only make to convert 37% of the jackpot amounts available into winnings. They lost

4 of the 11 jackpots they played for with the average lost jackpot equal to £17,320. Young

players fared slightly better in games versus mature players but the efficiency rate was still

less than 0.50.

The average efficiency rate of young versus mature games was high with only 9% of the

money lost. However, mature players were the main winners with an efficiency rate of 1.33.

In the 9 young versus mature games where there was a successful steal, six of the steals were

carried out by mature contestants and three by young contestants. The six mature contestants

stole an average of £22,460 off young contestants. By comparison, the three young

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contestants who managed a successful steal against a mature contestant won an average of

£16,467.

The efficiency rate in mature versus mature player games was exactly one, all of the available

jackpot money was won. Of the 14 jackpots, seven were split between the players and seven

were stolen.

Hair colour: Of the 43 females, 18 could be approximated as having blonde or fair hair with

25 being brunette or dark haired.4 Of the 18 blondes, 15 (or 83%) chose „split‟ while only 10

(or 40%) of brunettes chose „split‟. Blondes had a higher efficiency rate than brunettes.

Males cooperated with blondes 50% of the time (5 out 10 games) and with brunettes 54% of

the time (7 out of 13 games).

Table 7: Female hair colour and average game outcomes

Hair

Colour

Number Cooperation

Rate

Average

Jackpot

(Std. Dev.)

Average

Winnings

(Std. Dev.)

Average

Winnings /

½ Average

Jackpot

Blonde

or Fair

18 0.83 £15,271

(14,2001)

£6,376

(7,513)

0.84

Brunette

or Dark

25 0.40 £16,922

(20,089)

£6,279

(16,645)

0.74

Professions: To try and give an insight into the professions of those who chose „split‟ or

„steal‟ we can look at the 18 games that ended with a stolen jackpot. This gives us 18 stealers

and 18 suckers. Their professions are listed in table 8.

Of the two civil servants who played both chose „split‟ and both had „steal‟ played against

them. Other professions of those who had the jackpot stolen on them include; Storyteller,

Drama Tutor, Police Officer, Rtd Post Mistress, Hypnotherapist, Learning Support Worker,

Housewife, Actor, Receptionist and four from the marketing profession. A marketing

assistant, a marketing consultant, a marketing officer and an adverting executive all had a

jackpot stolen from them.

Among the professions of the successful stealers were; Car Dealer, Mortgage Broker, Sales

Assistant, Chef, Recruitment Consultant, Company Director, Café Owner and Tax

Consultant.

4 This is based purely on the observed rather than natural hair colour which may or may not be different.

14

Table 8: Professions of players involved in stolen jackpots (gender in brackets)

Stealers Suckers Jackpot

Marketing Manager (F) Learning Support Worker (M) £6,500

Area Manager (F) Storyteller (F) £47,250

IT Manager (F) Marketing Consultant (M) £7,710

Singer (M) Civil Servant (M) £3

Sales Assistant (F) Trainee Accountant (F) £20,220

Emergency Call Operator (F) Drama Tutor (M) £23,315

Train Conductor (M) Actor (M) £126

Car Dealer (M) Civil Servant (F) £50,500

Chef (M) Police Officer (F) £19,560

Student (M) Collection Agent (M) £1,815

Nurse (F) Housewife (F) £4,188

Teacher (M) Marketing Assistant (F) £16,600

Company Director (F) Advertising Executive (F) £66

Roofer (M) Hypnotherapist (F) £9,930

Recruitment Consultant (M) Rtd. Post Mistress (F) £17,400

Business Analyst (F) Project Co-ordinator (F) £4,100

Social Events Organiser (F) Account Executive (F) £61,060

Mortgage Broker (F) Office Manager (F) £30,185

6. Conclusion

The final part of the Goldenballs game show provides a natural experiment of a high stakes

prisoner‟s dilemma. In the episodes here the contestants play for over a half million pounds,

a figure which would be unattainable in a controlled experiment. Cooperation rates of close

to 50% are seen overall with some variation between groups. The identity of the opponent

has a role to play with less cooperation in games of the same gender and more cooperation

between players in the same age category. Overall, 17% of the money is left on the table

with mature players the most efficient at converting jackpots into winnings.

15

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a natural experiment in the prisoner's dilemma

Technology