Our first challenge involves a hypothetical decision-making problem, with no risk and only possible gain, if you choose correctly. The choice has you decide in favor of your reasoned self-interest, in light of what you think others will decide on theirs, and how they think others---including you---think they will decide, etc. Give this some thought, and then choose either Box A or Box B in your response.
Rules:
1. If you choose Box A, you will receive $1000, as long as everyone else chooses Box A as well; otherwise, nobody who chooses Box A will receive anything.
2. If you choose Box B, you will receive $100, as long as at least one-fourth of others choose Box B as well; otherwise, nobody who chooses Box B will receive anything.
RESULTS
Number of Respondents: 156
Number of votes for Box A: 134 (86%)
Number of votes for Box B: 22 (14%)
CONCLUSION: NO RESPONDENTS WIN ANYTHING! Since at least one responded to Box B, nobody who voted for A won anything; since not at least 1/4 voted for Box B, nobody who voted for B won anything.
A SUBMITTED ANALYSIS
Zeno's Coffeehouse Problem #1:
A Solution Via Maximizing Expected Utilities
Louis Marinoff
Department of Philosophy
The City College of New York
138th Street at Convent Avenue
New York, NY 10031
phone: (212) 650-7647
fax: (212) 650-7649
e-mail: marinoff@cnct.com
Contents:
1. The Problem
2. Why Maximize Expected Utilities?
3. Maximizing Expected Utilities
4. Assumptions in the Model
5. The Decision Calculus
6. A Sample Computation
7. A Sample Program
References
1. The Problem
You and a number of other players must each choose between box A,
which contains $1,000, and box B, which contains $100. If everyone
chooses box A, everyone receives $1,000; but if at least one player
chooses box B, then all those who choose box A receive nothing. If
one-fourth or more of all players choose box B, then each player who
chooses box B receives $100; but if less than one-fourth choose box B,
then all those who choose box B receive nothing.
2. Why Maximize Expected Utilities?
Some players do not need to be persuaded to maximize expected
utilities; these may skip to the next section. Other players may consider
maximizing expected utilities only as a last resort, having first sought (and
failed to find) alternative viable strategies.
This problem represents a many-player, non-zero-sum,
non-cooperative game with a twist. Like the prisoner's dilemma, it has
both a Pareto-efficient outcome (namely the case when all players choose
box A) and a Nash equilibrium (namely the case when more than
one-fourth of the players chooses box B). But unlike the prisoner's
dilemma, it lacks a dominant choice. That lack owes to the twist, which
obliterates dominance: the prevailing concern is the third possible
outcome, in which at least one but fewer than one-fourth of the players
choose box B, in which case no player receives any payoff. The game
matrix looks like this:
3. Maximizing Expected Utilities
The strategy of maximizing expected utilities is demonstrably robust
in some situations of risk or conflict of interestþ-notably in
non-cooperative games such as the prisoner's dilemma or Newcomb's
problem (e.g. Marinoff 1992, 1995a). The strategy is also useful in
situations in which the criterion of dominance fails to apply, such as in the
game under consideration here (see also Irvine 1993 and Marinoff 1995b,
1995c).
Although there is more than one way in which one can compute
expected utilities, we adopt herein the standard Savage formulation, in
which the expected utility of an outcome is the product of the probability
of that outcome and the value of that outcome in utiles (units of pure
utility). Hence, your expected utility of choosing box A is the product of
the probability that all other players choose box A and the value to you
of its contents. Similarly, your expected utility of choosing box B is the
product of the probability that at least one-fourth of the other players
choose box B and the value to you of its contents.
4. Assumptions in the Model
To render this proposed solution mathematically tractable and
computer programmable, we make the following standard assumptions.
First, the number of other players--n--is in theory arbitrarily large but
finite. In this treatment, however, we constrain n to a maximum of twenty
or so, in order to enable direct computation of the factorial terms entailed
by the algorithm. (The factorials of larger numbers can also be computed,
e.g. by Stirling's approximation.) Second, we assume that each of the
other players is predisposed to choose box B with some uniform known
probability p (which can range from zero to unity). Third, we assume that
the utilities of the boxes' contents are linear functions of the monetary
amounts. For utmost simplicity, let each utility equal the ordinal number
of dollars involved. Thus the utility of box A is 1,000 utiles;
that of box B, 100 utiles.
5. The Decision Calculus
For any given n (number of players) and p (probability that each
player will choose box B), we can compute the probability that exactly m
of n players will choose box B according to the binomial equation:
6. A Sample Computation
Suppose that there are twelve other players (n = 12). Tabled below
is a discrete spectrum of maximizing prescriptions that obtain as the
probability that each player chooses box B varies between zero and unity,
in increments of 0.1.
Table 1: Discrete Spectrum of Prescriptions for n=12
p-----p(0)--p(0 < m < n/4)--p(m > = n/4)--EUA-----EUB
0.0---1.00------.000---------------.000---------1,000-----0
0.1---.282------.607---------------.111---------282-------11
0.2---.069------.490---------------.442---------68.7------44.1
0.3---.014------.239---------------.747---------13.8------74.7
0.4---.002------.081---------------.917---------2.18-----91.7
0.5---.000------.019---------------.981---------0.24-----98.1
0.6---.000------.003---------------.997---------0.002----99.7
0.7---.000------.000--------------1.00----------0.0-------99.9
0.8---.000------.000--------------1.00----------0.0------100
0.9---.000------.000--------------1.00----------0.0------100
1.0---.000------.000--------------1.00----------0.0------100
The extrema of the spectrum yield self-evident results. When p equals
zero (which means that all players choose box A), the calculus assigns the
maximum and minimum possible expected utilities to boxes A and B
respectively (namely 1,000 and 0). When p equals unity (which means that
all players choose box B), the prescription is fully reversed.
The prescriptions become less intuitive--and therefore much more
useful--for intermediate values of p. For example, note that when p =
0.2, the most likely outcome is that at least one but less than one-fourth
of the players choose box B, in which case no-one would receive any
payoff. Even though it is more likely in turn that more than one-fourth of
the players choose box B than that no players choose box B, the
maximizing calculus still prescribes that we choose box A. This
prescription is conditioned by the relatively high payoff ratio A:B, among
other factors.
Empirically, for the given payoff structure, it is found that EUA =
EUB (i.e. the expected utilities of both choices are about equal) at
approximately p = .22, in which case we would be indifferent between
A and B. For all other values of p, the calculus of maximizing expected
utilities yields unequivocal prescriptions.
6. A Sample Program
Here is the annotated GW-BASIC code for the program that yields
the foregoing results. It prompts the user to input values for n (an integer
up to about 20) and p (a decimal between zero and one). For a given (n,p)
it outputs a probability distribution for every m between one and n, as
well as the row data that appear in table one.
05 REM Zeno's Coffeehouse Problem #1 Program
References
Irvine, I., 1993, How Braess' paradox solves Newcomb's problem,
International Studies in the Philosophy of Science 7, 141-60.
Marinoff, L., 1992, Maximizing expected utilities in the prisoner's
dilemma, Journal of Conflict Resolution 36, 183-216.
Marinoff, L., 1995a, The failure of success: intrafamilial exploitation in
the prisoner's dilemma, in: Danielson, P., ed., Modeling rational
and moral agents, Vancouver cognitive science series (Oxford
University Press), forthcoming.
Marinoff, L., 1995b, Probabilities and propensities in the prisoner's
dilemma, under review by the Journal of Theoretical Biology.
Marinoff, L., 1995c, The Cohen-Kelly queuing paradox revisited,
decongested and recongested, under review by the British Journal
for the Philosophy of Science.
10 REM This program computes the expected utilities of choosing box A
and box B.
20 REM It takes as input the probability P that each agent will choose box
B, assumed uniform over all agents.
30 REM It assumes twelve other agents, excluding yourself.
40 REM It assumes that the utility of money is a linear function of the
amount.
50 REM It outputs the probabilities that each possible number M agents
will choose box B, for M equals zero through twelve.
60 REM It outputs your expected utilities of choosing box A and box B.
100 INPUT "number of agents";N 'number of agents in addition to
yourself
110 INPUT "probability";P 'probability that an agent will choose box B
120 FOR M=0 TO N 'this loop computes the probability that M agents
will choose box B, for each possible M
130 J=N: GOSUB 280 'computes factorial N
140 NFAC=KFAC 'sets NFAC=factorial N
150 J=M: GOSUB 280 'computes factorial M
160 MFAC=KFAC 'sets MFAC=factorial M
170 J=N-M: GOSUB 280 'computes factorial N-M
180 OFAC=KFAC 'sets OFAC=factorial N-M
190 PM= (P^M)*(1-P)^(N-M)*(NFAC/(MFAC*OFAC)) 'computes
probability for given M
200 PRINT USING "##.## ";M;PM 'outputs M and associated
probability for that M
210 IF M=0 THEN PA=PM 'sets PA = probability that all agents will
choose box A
220 IF M>0 AND M
240 NEXT 'next value of M
250 PRINT USING ".### ";PA;PZ;PB 'outputs values of PA, PZ, PB
260 EUA=1000*PA: EUB=100*PB 'defines expected utilities of
choosing boxes A and B respectively
270 PRINT USING "###.## ";EUA;EUB 'outputs expected utilities
275 END 'end program
280 KFAC=1 'this subroutine computes factorial J
290 FOR K=1 TO J
300 KFAC=KFAC*K
310 NEXT
320 RETURN 'end subroutine