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The Dowry Puzzle
by Felix Baerlocher
WHAT IS IT?
The Dowry Problem
(Felix Baerlocher, Biology, Mt. Allison Unversity, Sackville, N.B., Canada; firstname.lastname@example.org)
This is a well-known puzzle from statistics (e.g., Mosteller 1965). A sultan wishes to test the intelligence of his advisor, who is seeking a wife. The sultan provides a choice of 100 women, each with a different dowry. The advisor has to choose the woman with the largest dowry. He is introduced to one woman at a time, who will tell him how much dowry she will be provided with, and he can either accept them or reject her. However, once he rejects a woman there is no going back, and he has to move on to the next candidate. Which strategy is most likely to provide him with the highest dowry?
The puzzle is also known as 'the secretary problem': an employer sequentially interviews a number of applicants trying to find the best. Rejection of an applicant is final (they might have accepted another job). Or, it could represent foraging animals: they encounter food patches containing variable amounts of food. Once they reject a patch, there is no turning back (another consumer might have found the food).
As it turns out, the highest probability of finding the top prize is the following strategy: examine the first 37 choices (or, more generally, n/e, where n = number of choices; Mosteller 1965). With this strategy, one finds the top choice in 37% of all cases.
Investigations of search or decision strategies, where options are encountered in a temporal sequence, have attracted the attention of statisticians, economists and biologists.
Statisticians in general have concentrated on finding the solution that is most likely to yield the top price (n/e rule).
Economists, in the context of job search (best salaries) and consumer choice (lowest prices), take into account search costs. They do not assume that only the 'best' solution is acceptable. They may set a 'reservation price' where the marginal cost of further search equals marginal expected improvement over the best prospect seen so far (these models generally assume that previously seen options are still available).
Biologists tend to emphasize empirical testability of their models. They are generally more aware of search and opportunity costs (moving from one place to the next takes time, during which no feeding takes place), and handling costs (how much energy has to be invested to subdue, handle, digest the prey). One popular model has been a 'reservation price' rule: by sampling the environment, the animal constructs an expectation of what might reasonably be expected. As soon as this value is reached, the reward (food, mate) is accepted. Simon (1990) calls this a 'satisficing' model. It is a form of 'bounded rationality', useful in situations where the entire set of possible options is not known.
HOW IT WORKS
The graphics window is changed to show a racetrack, with Xcor between -50 and +50, for a total of 101 patches.
There are two turtles, a shuffler and a player. The shuffler has numbers from 1 to 100, which he randomly distributes on patches -49 to +50 (one number per patch).
The player moves forward, one patch at a time, and makes a list of all numbers he encounters. After a number of steps (can be modified by slider), he determines the highest values found so far (standard). He then continues to move forward until he finds a patch with a higher number, or until he runs out of options (last patch). One of these values is the reward, which is placed in the rewards list. The process is repeated a number of times (set by cycles slider).
The program keeps track of 100s (number of times that the maximum is chosen), average (average payoff) and med (median of payoffs). It also keeps track of moves (number of steps; this combines the initial search steps and steps taken until a reward is accepted).
HOW TO USE IT
Setup: prepares the model for the first run (shuffler distributes 100 values; player at end of lane).
Go: initiates search behaviour
Steps: slider, how many patches are examined to set a standard
Cycles: slider, how many times does the playe run through the lane and decide on a reward
moves: how many moves does the player make before he accepts a reward. Dividing moves by cycles gives average moves per decision
hundreds: how many times has the top reward been found?
average: what is the average reward?
med: what is the median reward?
new-shuffle?: redistributes awards with each new cycle
THINGS TO NOTICE
Running averages, med and number of 100s shown.
THINGS TO TRY
Check if the 37 rule is indeed true. Vary the number of steps and determine Average, Med, 100s.Try Behavior space to observe changes in Average, Med, 100s as a function of steps.
EXTENDING THE MODEL
For a more realistic approach to optimal search strategies, quantify costs for search behavior and benefits for the reward. Differentiate between costs of simply inspecting a patch (initial phase) and exploiting a patch (in biology, this could include costs of overpowering a prey, eating and digesting the food).
If the model represents mate choice, modify it so that both mates have a choice.
How does the payoff (optimal choice) change if the player can choose previously encountered patches?
What if the energy level of player is limited, and your primary objective is to prevent starvation? Is the optimal strategy the same as the one that gives the highest average payoff?
Or, the primary objective can be to be reproductively as successful as possible (highest numbers of offspring per unit time, Darwinian fitness). It is always best to maximize life-span?
Food is usually distributed randomly. This could be simulated by using standard square arrangement of patches, and randomly distribute 100 patches with values of 1 to 100. Or, some of the numbers between 1 and 100 could be replaced by 0. What would the optimal search strategy be now?
Adding more players: this would introduce an element of competition.
Some patches might be provided with a very high return, but may also be dangerous (predators). Might it be more advantageous to avoid these high-yield patches? In Biology, the concept of 'enemy-free' space plays a large role when discussing optimal foraging.
The model is still preliminary; I'm not entirely convinced that all steps work as they should. I also think it can probably be simplified.
Anything related to consumers, herbivores, predator-prey interactions
CREDITS AND REFERENCES
Mosteller, F. (1965) Fifty challenging problems in probability with solutions. Dover, New York.
Simon, H.A. (1990) Invariants of human behaviour. Annual Review of Psychology 41, 1-19.
Todd, P.M. & Miller, G.F. (1999) From pride and prejudice to persuasion - satisficing in mate search. Pp. 287 - 308 in Gigerenzer, G., Todd, P.M. and the ABC Research Group (eds.), Simple heuristics that make us smart. Oxford University Press.
I benefitted greatly from attending the Workshop (Mesa State College & CCL, July 2003, by Dr. Michael Gizzi and colleagues). I also received numerous tips from Seth Tisue and other members of the netlogo user group, for which I'm extremely grateful.