I am sure there are many ways to analyze this situation, but the following one is appealing for its simplicity and use of only the most basic properties of probability. Assuming ties for heads cause the game to continue until a result is uniquely determined, it shows the chance of a player winning the game is her relative odds: the proportion of her odds of heads, divided by the sum of the odds of heads among all players. The game length obviously has a geometric distribution. Its parameter is proportional to this sum of odds. The constant of proportionality is the product of all the chances of tails.
Other methods of resolving ties (to create a definite winner) can be analyzed in a similar way, starting with computing the chance that a given player will win a particular round and continuing as shown here.
Let there be $m$ players, each using a coin with probability $p_i$ of heads. Let $i,j,\ldots, k$ be any permutation of $(1,2,\ldots,m)$.
Suppose that when a tie occurs, no outcome is declared and play continues. Then the chance that player $i$ wins is the chance that she is the sole person to toss heads, equal to
$$p_i(1-p_j)\ldots(1-p_k) = \frac{p_i}{1-p_i}\prod_{l=1}^m (1-p_l) = \pi_i Q$$
where I have written $\pi_i = p_i/(1-p_i)$ for player $i$'s odds of heads and $Q$ for the product of all $1-p_l$ (the chance that everybody simultaneously observes a tail).
If no player wins a round, the game starts over, with exactly the same probabilities. Therefore the chance that player $i$ wins the entire game is the chance they can win a given round, divided by the sum of all players' chances to win the round:
$$\Pr(i\text{ wins}) = \frac{\pi_i Q}{\sum_{l=1}^m \pi_l Q} = \frac{\pi_i}{\sum_{l=1}^m \pi_l} = \frac{\pi_i}{\pi}.$$
It is their proportion of the total odds $\pi$.
The chance that the game ends on any particular round is the chance that exactly one player observes heads, equal to
$$\sum_{l=1}^m \pi_l Q = \pi Q.$$
Thus the length of the game has a geometric distribution with parameter $\pi Q$.
This result is supported by simulation:
It was carried out with this R script, which can simulate and summarize tens of millions of tosses per second for arbitrarily many players (up to limits determined by RAM).
coins <- log(2:10); coins <- coins / sum(coins)
n.players <- length(coins)
n.rounds <- 1e6
#
# Simulate many rounds.
#
system.time({
tosses <- matrix(runif(n.rounds * n.players), nrow=n.players) < coins
wins <- colSums(tosses) == 1
winners <- colSums(1:n.players * tosses) * wins
(results <- tabulate(winners[winners != 0], n.players))
})
#
# Compare to theory.
#
odds <- coins / (1-coins)
p <- odds / sum(odds)
rbind(Simulation=results / sum(results), Theory=p)
#
# Display the distribution of game lengths compared to the geometric distribution.
#
lengths <- diff(c(0, which(wins==TRUE)))
Q <- prod(1 - coins)
pQ <- sum(odds) * Q
n.max <- ceiling((log(.002) + log(pQ)) / log(1-pQ)) # Plotting limit
prob.geometric <- (1 - pQ)^(1:n.max - 1) * pQ
subtitle <- paste(sum(results),"games played with coins",
paste(round(coins, 2), collapse=","))
hist(lengths[lengths < n.max], freq=FALSE, breaks=(1:n.max)-1/2,
main="Simulation (bars) vs. Theory (lines)",
xlab="Game lengths",
sub=subtitle)
invisible(sapply(1:n.max, function(i) {
lines(c(i,i), c(0, prob.geometric[i]), lwd=2, col="Red")
}))
