Friend or Foe:

a real prisoner's dilemma

In 2002 the Game Show Network ran a show called Friends or Foe, hosted by former MTV VJ, Kennedy. In the show the players were paired together and answered trivia questions as a team. The winning team got to go home with all the cash. BUT.

Before the winning team walked away with their winnings they faced one last challenge called the trust box. In this part of the game, the two members of the winning team had a simple choice FRIEND or FOE.

If both choose FRIEND then the split their winnings evenly. However, if one of the contestants chooses FOE then that contestant gets all the winnings to themself and their teammate gets nothing. If they both choose FOE then they both walk away with nothing.

A number of other game shows have also used the Prisoner's Dilemma. Here is a link to the Australian comedy game show, Guy Montgomery's Spelling Bee: https://youtube.com/shorts/0LmXzvNjrwQ?si=aYaL0f_iwFsnMz00.

The British game show Golden Balls (yep) similarly finished with a prisoner's dilemma. In this show the choices are SPLIT or STEAL, where both players split the winnings if they both choose SPLIT. If a contestant chooses STEAL they get all the winnings. But if they both choose STEAL they get nothing.

The Game

Click on and watch the youtube video with Tony and Lucy playing the Split or Steal game. What is the formal representation of that game? Who are the players? What are their strategies? What are the payoffs for each outcome (set of strategies)?

In the video the players have time to discuss their strategies. But this discussion is not represented in the game below. Should it be?

Players: Tony and Lucy

Strategies: SPLIT or STEAL

Payoffs:

SPLIT, SPLIT: 33, 442.50 each

SPLIT, STEAL: Tony gets nothing and Lucy gets 66,885

STEAL, SPLIT: Tony gets 66,885 and Lucy gets nothing.

STEAL, STEAL: 0 each

Below is a picture of a generic Prisoner's Dilemma game. We call this the normal form representation. Can you create a normal form for the Split or Steal game?

(Weakly) Dominant Strategy

STEAL is a (weakly) dominant strategy. If Lucy chooses SPLIT then Tony is better off choosing STEAL by 33,442.50. If Lucy chooses STEAL then Tony gets nothing either way so he might as well choose STEAL. Similarly for Lucy, STEAL is a (weakly) dominant strategy.

Equilibrium

STEAL, STEAL is a weakly dominant strategy equilibrium of the game.

Is it Pareto optimal? No. Both players can be made better choosing another outcome, {SPLIT, SPLIT}.

The host makes the joke that the network thanks the contestants for donating their money. Obviously, the point of the game is to keep the money for the network. Could you design a game that would increase the probability that both players choose Steal?

Friend or Foe Data

A table (R code)

library(Ecdat)

library(tidyverse)

data("FriendFoe")

game_mat = FriendFoe |>

summarize(

mean(play == "friend" & play1 == "friend"),

# this creates a vector of 0 and 1s depending whether

# the statement inside the parenthesis is true

# note the use of "==" to ask whether something is true

mean(play == "foe" & play1 == "friend"),

mean(play == "friend" & play1 == "foe"),

mean(play == "foe" & play1 == "foe")

) |>

matrix(, nrow = 2)

colnames(game_mat) = rownames(game_mat) = c("friend","foe")

What does the table show? Which outcome is most likely to occur in the game? Are players more likely to pick FRIEND or FOE?

Reshaping the data (R code)

FriendFoe = FriendFoe |>

mutate(

similar = (sex == sex1) + (max(abs(age - age1)) - abs(age - age1))/10 + (white == white1)

)

df1 = FriendFoe[,c(1:4,12)]

df2 = FriendFoe[,c(8:11,13)]

colnames(df2) = colnames(df1)

df = rbind(df1, df2)

df$game = rep(1:227, 2)

df$round = rep(FriendFoe$round, 2)

df$season = rep(FriendFoe$season, 2)

df$cash = rep(FriendFoe$cash, 2)

df$similar = rep(FriendFoe$similar, 2)

The code reshapes the data so that each player is an observation. It also creates a measure of similarity between the players.

Regressions (R code)

lm1 = lm(cash ~ sex + white + age + round + season, data = df)

glm1 = glm(play == "friend" ~ sex + white + age + similar + cash,

family = binomial(link = "logit"), data = df)

glm2 = glm(play == "friend" ~ sex + white + age + similar + cash,

family = binomial(link = "probit"), data = df)

What observable characteristics determine the choice people make? Does the size of the prize matter? Does gender or race or age matter? Does it matter how similar the contestants are?

Nick's Strategy

One episode of Golden Balls was a little different form the rest. While it was often the case that contestants would try to convince each other that they were lovely people and would never choose STEAL, Nick decided to say that he would definitely STEAL.

Nick's strategy was to convince Ibrahim that Nick was definitely going to choose STEAL and that Ibrahim should choose SPLIT.

As above, does this discussion matter?

Interviews with both Nick Corrigan and Ibrahim Hussein on RadioLab.

Does Nick change the game? Can you write down the new game? What is the equilibrium of the new game?