I did not solve the problem analytically but I performed a simulation with 100 different $a/b$ ratios varying from 0.01 to 1. $a$ is the number of dice of player A and $b$ is the number of dice of player $b$. For each ratio I simulated 1000 games and computed the multiplicative constant.

This what I got:

For the dice I assumed a uniform distribution between 0 and 1.

I did not solve the problem analytically but I performed a simulation with 100 different $a/b$ ratios varying from 0.01 to 1. $a$ is the number of dice of player A and $b$ is the number of dice of player $b$. For each ratio I simulated 1000 games and computed the multiplicative constant.

This what I got:

For the dice I assumed a uniform distribution between 0 and 1.

If we take the same ratio the expected value for the multiplicative constant is the same. I tested with a ratio of $0.5$ timing $a$ and $b$ up to a factor of 2000. Here the results as scatter plot and density distribution

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