Some thoughts on combining Martingales and Dollar Auction

This is a little ditty to convince you (and me too) that we might not be wasting our time if we wanted to pursue thinking about the following kind of game.

Setup

Consider this: A Game is joined between a single punter (henceforth, the Trump Player) and the House. The Trump player does not use his own money, but is backed by a series of other players. The Trump player is playing the Martingale, a betting scheme where at the outcome of each gamble, he plays ‘Double or Nothing’. It is a straightforward calculation (see Grimmett and Stirzaker or similar) to demonstrate that regardless of the probability of win, \(p\), the Trump player will eventually win, and will gain a profit of 1 unit, because \(2^N - \sum_{i=0}^{N-1} 2^n = 1\) The Trump player is able to exercise optional stopping; we call this the Trump player’s strategy.

The Trump player does not use his own money, but rather relies on his backers, each of whom provides an equal share of the next round’s bet; for example, if there are B backers, then in the \(N^{th}\) round each will supply \(\frac{2^N}{B}\) units, which also happens to be their investment to date. Because the Backers will each share in the eventual, assured payoff, continuing the game is rational (in a sense, which is the exploration of this note) and should they endure, they will receive a payoff of \(\frac{1}{B+1}\), with a single share going to the Trump player.

While we note that the eventual payoff is assured, there is a rub; it can also be demonstrated that the expected losses for the Trump Player before the eventual win are infinite for LOOKUP some probability condition on p.

At each round of the game, backers are able to make a single, irrevocable decision to quit the scheme. This unilateral decision affects all of the players in the game, increasing both the eventual payoff and required investment for the current and all subsequent rounds. The backers have finite resources that are not necessarily identical for all players. The backers’ risk tolerance is a function of their own resources, for which they have perfect information, and their presumption of the other backer’s resources, for which they have imperfect information. No player knows the true value of \(p\), except that it is not identically zero.

The static ‘house’, against whom the game is played, will continue the game until the Trump Player stops or runs out of resources. It is assumed to have infinite resources. The Trump player has no resources of their own and will continue to play so long as there are resources.

Subjective Probability.

Neither the backers nor the Trump Player know the true value of \(p\) and will need to use their estimate, which we call \(\hat{p}\). As a simple device for exploring the problem, we presume that in round \(N\) the subjective probability is \(\frac{1}{1+N}\) NOTE the right scaling might be \(\frac{1}{N^2}\) or whatever makes the math work out

, so that in the first round, the probability is 1/2, in the second round 1/3, and so on.

Scenario 1: A single backer with initial resource \(S\).

A single backer should continue with the Trump Scheme so long as \(\frac{1}{2} \prod_{n=1}^{N} (n+1)^{-1} > (1-(n+1)^{-1} 2^N\),

Note, I think this is wrong, but there will be some condition like this

Which is to say that the game should be played for the single backer so long as the discounted future value of the payoff is greater than discounted risk.

This is the jumping off point

Under what conditions would anyone play this game?