3.1 Fair Games
Sequence of fair games, gambling strategy.
Consider a gambler, who can bet on a sequence of fair games. Specifically, there is a sequence \(X_1, X_2, X_3, \dots\) of games such that if the gambler chooses to bet an amount \(\alpha \in \mathbb{R}\) at time \(n\), then the gambler receives the random amount \(\alpha X_{n+1}\) (including the stake, if returned) at time \(n+1\). For example, in a game of Roulette, betting on a single number, then \(X_n = 36\) with probability \(1/36\), and \(X_n = 0\) (lose the stake) otherwise, each game being i.i.d.. We call the sequence fair if \(X_n \in \mathcal{L}^1\),4 and the average winnings, given the previous outcomes, is equal to the amount bet, i.e. for any \(\alpha \in \mathbb{R}\): \[\mathbb{E}\left[ \alpha X_n |X_1, X_{2}, \dots, X_{n-1} \right] = \alpha \implies \mathbb{E}\left[ X_n | X_{1}, X_{2}, \dots, X_{n-1} \right] = 1.\] In particular, assuming we start with wealth \(\beta\) at time \(n-1\) and bet \(\alpha\) on the game, our new wealth will be \(\beta + \alpha(X_n-1)\), since we lose the stake \(\alpha\) which we bet.
Note that we do not generally require the games to be independent. For example, consider betting on whether Bath Rugby win each of their matches in a season. Their chances of winning the fifth match of the season will change as we observe how the previous games go. The key requirement for being a fair game is that, at the point where we decide to bet on the fifth game (after the first four have happened), the odds we receive will depend on what has happened in the first four games.
Definition 21. Suppose \((X_n)_{n \ge 1}\) is a sequence of fair games. A gambling strategy \((\alpha_n)_{n \ge 1}\) is a sequence of functions, \(\alpha_n = \alpha_n(X_1, X_2, \dots, X_{n-1})\) depending on the previous games, which represents the amount wagered on the \(n^{\text{th}}\) game. (Note that we will often write simply \(\alpha_n\) for \(\alpha_n(X_1, X_2, \dots, X_{n-1})\).) A bounded gambling strategy is a gambling strategy where there exists \(K>0\) such that \(|\alpha_n(x_1,x_2,\dots,x_{n-1})| \le K\) for all \(n\) and all \(x_1,\dots, x_{n-1}\).
The gains process \((G_n)_{n \ge 0}\) associated with a particular gambling strategy \(\alpha_n\), and the sequence of fair games \((X_n)_{n \ge 1}\) is the total winnings after the first \(n\) games, so: \[G_0 = 0, G_n = G_{n-1} + \alpha_n(X_1, X_2, \dots, X_{n-1}) (X_n-1).\] In particular, \[G_n = \sum_{k = 1}^{n} \alpha_k(X_1, X_2, \dots, X_{k-1}) (X_k-1).\]
Lemma 22. Let \(G_n\) be the gains process associated with a bounded gambling strategy \(\alpha_n\) and a sequence of fair games \(X_n\). Then the average gain, \(G_n\), satisfies: \[\begin{align} \tag{3.1} G_n = \mathbb{E}\left[ G_{n+1} | X_1, X_2, \dots, X_n \right],\end{align}\] and \(\mathbb{E}\left[ G_n \right] = G_0 = 0\) for all \(n\).
Note that the condition (3.1) tells us that my expected winnings after one more fair game is the same as my current winnings. Since this is true for any (bounded) gambling strategy, the result tells us that we cannot find a ‘clever’ gambling strategy that will allow us to beat a fair game.
We have to consider bounded gambling strategies here, since we need to ensure the conditional expectation is well defined. Recall that in Definition16, we needed our random variable to be integrable in order to define conditional expectation.
Solution:
We have \[\begin{align} % G_0 =& 0\\ G_{n+1} =& G_{n} + \alpha_{n+1}(X_1, X_2, \dots, X_{n}) (X_{n+1}-1).\end{align}\] So (by linearity of conditional expectation): \[\begin{align} \mathbb{E}\left[ G_{n+1} | X_1, X_2, \dots, X_n \right] = & \mathbb{E}\left[ G_{n}| X_1, X_2, \dots, X_n \right]\\ {} & + \mathbb{E}\left[ \alpha_{n+1}(X_1, X_2, \dots, X_{n}) (X_{n+1}-1) | X_1, X_2, \dots, X_n \right]\end{align}\] But by the expression just before the lemma, \(G_n\) is a function of \(X_1, \dots, X_n\), so by t.o.w.i.k. \[\begin{align} \mathbb{E}\left[ G_{n+1} | X_1, X_2, \dots, X_n \right] =&G_{n}+\alpha_{n+1}(X_1, X_2, \dots, X_{n}) \mathbb{E}\left[ (X_{n+1}-1) | X_1, X_2, \dots, X_n \right]\end{align}\] Since \(X_{n+1}\) is fair, \[\mathbb{E}\left[ X_{n+1} | X_1, X_2, \dots, X_n \right]=1 \implies \mathbb{E}\left[ X_{n+1} -1 | X_1, X_2, \dots, X_n \right]=0\] and we conclude \[\begin{align} \mathbb{E}\left[ G_{n+1} | X_1, X_2, \dots, X_n \right]=G_{n}\end{align}\] as required.
Now use tower property again \[\begin{align} \mathbb{E}\left[ G_{n+1} \right] =&\mathbb{E}\left[ \mathbb{E}\left[ G_{n+1} | X_1, X_2, \dots, X_n \right] \right] = \mathbb{E}\left[ G_n \right] = \mathbb{E}\left[ G_{n-1} \right] =\dots= \mathbb{E}\left[ G_0 \right]=0.\end{align}\]
\(\square\)
Recall that this means that \(\mathbb{E}\left[ |X_n| \right] < \infty\) for each \(n\).↩︎