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Generalizations of the M&M Game

Snehesh Das, Evan Li, Steven J. Miller, Andrew Mou, Geremias Polanco, Wang Xiaochen, April Yang, Chris Yao

TL;DR

The paper generalizes the classical two-player M&M Game to derive finite, exact formulas for the expected number of turns and extends the model to multiple coins and evolving head-probabilities. It uses memoryless-process arguments, geometric waiting-time decompositions, and combinatorial sums to obtain precise results, including a symmetry theorem for dual coin-value configurations and a finite expression for the standard game's expectation. It then models dynamic depletion with an exponential-heads probability and fits a Gompertz curve to the tie probability, supported by simulations that show the depletion-rate parameter $\lambda$ dominates over the initial count $n$ in determining outcomes. The work suggests multiple avenues for future research, such as time-dependent probabilities, loop-erased dynamics, multiplayer extensions, and alternative distributions, with practical implications for educational probability games and probabilistic depletion processes.

Abstract

The M&M Game was created to help young kids explore probability by modeling a response to the question: \emph{If two people are born on the same day, will they die on the same day?} Each player starts with a fixed number of M&M's and a fair coin; a turn consists of players simultaneously tossing their coin and eating an M&M only if the toss is a head, with a person ``dying'' when they have eaten their stash. The probability of a tie can naturally be written as an infinite sum of binomial products, and can be reformulated into a finite calculation using memoryless processes, recursion theory, or graph-theoretic techniques, highlighting its value as an educational game. We analyze several extensions, such as tossing multiple coins with varying probabilities and evolving probability distributions for coin flips. We derive formulas for the expected length of the game and the probability of a tie by modeling the number of rounds as a sum of geometric waiting times.

Generalizations of the M&M Game

TL;DR

The paper generalizes the classical two-player M&M Game to derive finite, exact formulas for the expected number of turns and extends the model to multiple coins and evolving head-probabilities. It uses memoryless-process arguments, geometric waiting-time decompositions, and combinatorial sums to obtain precise results, including a symmetry theorem for dual coin-value configurations and a finite expression for the standard game's expectation. It then models dynamic depletion with an exponential-heads probability and fits a Gompertz curve to the tie probability, supported by simulations that show the depletion-rate parameter dominates over the initial count in determining outcomes. The work suggests multiple avenues for future research, such as time-dependent probabilities, loop-erased dynamics, multiplayer extensions, and alternative distributions, with practical implications for educational probability games and probabilistic depletion processes.

Abstract

The M&M Game was created to help young kids explore probability by modeling a response to the question: \emph{If two people are born on the same day, will they die on the same day?} Each player starts with a fixed number of M&M's and a fair coin; a turn consists of players simultaneously tossing their coin and eating an M&M only if the toss is a head, with a person ``dying'' when they have eaten their stash. The probability of a tie can naturally be written as an infinite sum of binomial products, and can be reformulated into a finite calculation using memoryless processes, recursion theory, or graph-theoretic techniques, highlighting its value as an educational game. We analyze several extensions, such as tossing multiple coins with varying probabilities and evolving probability distributions for coin flips. We derive formulas for the expected length of the game and the probability of a tie by modeling the number of rounds as a sum of geometric waiting times.

Paper Structure

This paper contains 12 sections, 6 theorems, 30 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Expected Number of Turns of the M&M Game
  3. Multiple Coins with Different Values
  4. Evolving Coin Probabilities: an Exponential Distribution
  5. Simulation Results
  6. Proposed Model: Gompertz Curve Fit
  7. Impact of $\lambda$
  8. Conclusion and Future Work
  9. Acknowledgments
  10. Code for the expected number of turns
  11. Code for the Several Coins Game
  12. Code for Evolving Coin Probabilities

Key Result

Theorem 1.1

The probability the M&M Game ends in a tie with two people using fair coins and starting with $k$ M&M's is

Figures (5)

  • Figure 1: Probability of a tie for varying $k$ values.
  • Figure 2: Probability of a tie as a function of $p$, with different number of M&M's
  • Figure 3: This graph shows the idea of the proof, each player starting with $k$ M&M's, the set of coins are (3,-2,-1). The trajectories represent their M&M's in each round. As long as the penultimate round (in yellow) hits the range of a box with side length $3$ and both players eat all their remaining M&M's in the last round, there will be a tie.
  • Figure 4: Probability of a tie with different $\lambda$ values.
  • Figure 5: Gompertz fit of $\lambda$ values on the probability of a tie.

Theorems & Definitions (8)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 1.2
  • Theorem 2.1
  • Theorem 3.1
  • proof
  • Conjecture 4.1