Optimal Play in Sequential Gladiator Duels with Strength-Based Win Odds

Summary

This document poses a sequential elimination game between two teams of four gladiators. In each round, Alice chooses a fighter first and Bob responds with a matchup; each duel has no tie, and the stronger fighter’s advantage is expressed as a probability based on the two fighters’ strengths. A winner keeps fighting while a loser is removed, so choices affect both the immediate duel and the matchups available later.

The central question is the probability Bob wins when both players choose optimally. The post asks whether exponential clocks or random-order models might reveal a solution, but reports no derivation, computed probability, or evidence that resolves the game. Its value is as a compact stochastic-game problem involving sequential choices, state-dependent outcomes, and optimal strategy. Any answer would need to account for all remaining fighters and the order of choice; the document itself leaves that analysis open.

Key ideas

• The game consists of sequential duels with fighters eliminated after losing.
• Alice chooses first in each round, and Bob then selects the opposing fighter.
• Each duel’s win probability depends on the strengths of both gladiators.
• A surviving victor retains their strength for later rounds.
• The document asks for Bob’s optimal win probability but does not provide a solution.

Tags

# Optimal Strategy in Sequential Gladiator Duels with Probabilistic Outcomes, Colosseum Fight II


# Optimal Strategy in Sequential Gladiator Duels with Probabilistic Outcomes, Colosseum Fight II












I did find this problem on https://www.quantguide.io/questions/colosseum-fight-ii which is as follows:

Alice and Bob are in ancient Roman times, each commanding a team of 4 gladiators. Alice’s gladiators have strengths of $1, 2, 3,$ and $4,$ while Bob’s gladiators have strengths of $4, 5, 9,$ and $12.$

The tournament will proceed as a series of one-on-one battles. In each round, Alice selects one of her remaining gladiators to fight, followed by Bob choosing one of his remaining gladiators to face Alice’s choice. The selected gladiators then battle to the death, with no possibility of a tie.

The probability that a gladiator with strength $x$ defeats one with strength $y$ is given by the formula: $$\frac{x}{(x + y)}.$$ The victor retains their strength and continues in future battles; the loser is eliminated.

The tournament continues until one player has no gladiators left. The winner is the player who still has at least one surviving gladiator.

Question: What is the probability that Bob wins the tournament, assuming both players play optimally?

I’ve tried embedding this into more familiar probabilistic models—such as exponential clocks or random ordering —but I haven’t found a breakthrough.

Any insight into the structure or potential methods would be greatly appreciated.

Thanks!

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.