Can Poker Chip Arrangements Be Optimized for Maximum Profit?

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Discussion Overview

The discussion revolves around optimizing the arrangement of poker chips to meet specific conditions: a total value of $40 and exactly 60 chips, while exploring various constraints and methods for finding a solution. The focus includes mathematical modeling, optimization techniques, and the implications of constraints on the problem.

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • One participant presents a scenario involving four denominations of poker chips and seeks to determine the optimal arrangement under given conditions.
  • Another participant questions the definition of "optimal" in this context, suggesting it is vague.
  • A participant notes the presence of two equations and four unknowns, discussing the potential for linear programming and the search for integer solutions.
  • Constraints such as x > y > z > w >= 0 are proposed, with a focus on maximizing the number of smaller chips.
  • There is a discussion about the nature of constraints and their role in optimization problems, with emphasis on the need for ranking solutions.
  • A related problem, the change-making problem, is mentioned as a reference for similar optimization challenges.
  • One participant expresses that the solution may be approached using VBA in Excel after reviewing relevant resources.

Areas of Agreement / Disagreement

Participants express differing views on what constitutes an "optimal" solution and the implications of constraints, indicating that multiple competing perspectives remain in the discussion.

Contextual Notes

Participants highlight the need for additional equations to solve the system and the implications of integer constraints, which may limit the solution space. The discussion also reflects on the ranking of solutions based on defined criteria.

whdahl
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Hey everyone. I was pondering how best to optimize a chip arrangement for a poker game. This is the scenario I've thought up:

There are 4 denominations of colored chips with a set value.
White (W) = 0.05
Red (R) = 0.25
Blue (B) = 1.00
Green (G) = 5.00

A player wants to purchase 40 dollars worth of chips. If he must receive exactly 60 chips total, what is the optimal amount of each chip denomination to give the player?

These two conditions, (the $40 buy in and the 60 chip amount) will yield two equations:

xW + yR + zB + wG = 40
x + y + z + w = 60

There are 2 equations and 4 unknowns. Where might I find two more equations so that I can solve the equations, or is there some method using calculus that would yield a result?
 
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What do you consider to be an "optimal" amount? This is a very vague term in this context.
 
Last edited:
whdahl said:
There are 2 equations and 4 unknowns. Where might I find two more equations so that I can solve the equations, or is there some method using calculus that would yield a result?
You also have the constraints that x>=0, y>=0, z>=0 and w>=0 and that x, y, z, w are integers. And you have your expressed desire for optimality.

In the absence of a constraint to integer values this might be an exercise in "linear programming" -- find the maximum of a linear function in n real-valued variables given a set of linear inequalities that those variables must satisfy.

With the restriction to integer values, this is at worst a matter of searching a finite number of possibilities for an optimum.
 
Another constraint would be that x > y > z > w >= 0 because it is best to have the highest amount of smaller chips. Is there an efficient way, using matrices perhaps, of solving for solutions?
 
whdahl said:
Another constraint would be that x > y > z > w >= 0 because it is best to have the highest amount of smaller chips. Is there an efficient way, using matrices perhaps, of solving for solutions?
If your job were to hand out $42 using exactly 10 chips, would this "constraint" still apply?

When describing optimization problems, a "constraint" is a hard requirement which must be met. Any possible solution must meet each and every constraint. In addition to the constraints, you generally have a way to rank the possible solutions to see which one(s) are best. A solution which is tied for best is "optimal".

"x > y > z > w >= 0" has the form of a constraint. It does not provide a way to rank solutions except in the crudest of ways (all solutions which satisfy the inequality are tied for best).

You have said that "it is best to have the highest amount of smaller chips". One possibility is that you want the solution that gives the player the highest possible number of white $0.05 chips. If multiple solutions maximize the number of white chips you want the solution that gives the player the highest possible number of red $0.25 chips. And so on. Is that what you are after?

A related problem is: https://en.wikipedia.org/wiki/Change-making_problem
 
jbriggs444 said:
You have said that "it is best to have the highest amount of smaller chips". One possibility is that you want the solution that gives the player the highest possible number of white $0.05 chips. If multiple solutions maximize the number of white chips you want the solution that gives the player the highest possible number of red $0.25 chips. And so on. Is that what you are after?

Yes.
After reading through that wiki page and another on dynamic programming, it seems that this is a problem that can be solved using VBA in excel.
 

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