Proebsting’s Paradox: Kelly Betting and Changes in Wealth

# Proebsting's Paradox - Wealth Effect of Changing Odds - Kelly Criterion


# Proebsting's Paradox - Wealth Effect of Changing Odds - Kelly Criterion












I'm reading Volatility Trading by E. Sinclair and at pag. 154 I've found this paradox. It is addressed also in Zambrano (2014). But its solution is not clear to me. Basically it revolves around:

> What is the change in the gambler's, a Kelly bettor, bankroll when the odds move from 2/1 to 5/1? This is the value $\gamma$ that satisfies: $$ 2-\gamma = 5 (1+\gamma) $$ that is, entering a 2/1 bet for $.25$ of his bankroll $W$ and then having the odds change to 5/1 is the same as not having bet, having the bankroll change by $-.5\times.25W$ and then betting at 5/1 odds.

Can anyone explain this to me and how is related to the equation above?

Thanks. Let me know if more details are needed.

## Answer by Achrbot (score 1, accepted)

https://quant.stackexchange.com/a/81758

A Kelly better wants to maximize the expected log utility of wealth $\mathbb{E}\left[\log W_T\right]$ where $W_T$ is final wealth.

Now consider a bet paying out $W_0f(1+b)$ with probability $p$ for each fraction $f$ of initial wealth $W_0$ at stake. The Kelly optimal fraction $f^*$ for such a one-time bet, is given by the Kelly formula $$ f^* = p - \frac{1-p}{b} $$

For example, if $p=0.5$ and $b=2$, then $f^* = 0.25$.

Now consider the following scenario:

- A Kelly agent has entered the above bet, waging $f^*=0.25$ at 2-1 odds.

- After placing the bet, they are now offered to place an additional wager on the same outcome, but at 5-1 odds.

We can find the optimal bet on the second wager $f_2^*$, by maximizing \begin{align*} \mathbb{E}\left[\log W_T\right] &= 0.5\log\left(W_0(1+0.25\cdot 2 + 5f_2\right)) \\ &\phantom{=}+ 0.5\log\left(W_0(1-0.25-f_2)\right) \end{align*} and get $f^*_2 = 0.225$, with total utility \begin{align} \mathbb{E} \left[\log W_T\right] &= 0.5\log \left(2.625\cdot 0.525 \right) + \log W_0 \end{align}

Proebsting's "paradox" is that the total fraction at stake, is $0.25 + 0.225=0.475$, which is higher than the $0.4$ which would have been wagered, if the agent had only been offered the second bet. This is in spite of the lower average odds received by the agent.

To resolve the paradox, consider another agent with initial wealth $\tilde{W}_0$, that is only offered the second bet. They will wager $\tilde{f}^* = 0.4$, getting expected utility \begin{align*} \mathbb{E}\left[\log \tilde{W}_T\right] &= 0.5\log(\tilde{W}_0(1+5\cdot0.4)) + 0.5 \log(\tilde{W}_0(1-0.4)) \\ &=0.5\log(3\cdot 0.6) + \log\tilde{W}_0 \end{align*} We can now answer the question, which levels of initial wealth, would make both agents equally well off? Setting both utilities equal, we get \begin{align*} 0.5\log (2.625 \cdot 0.525) + \log W_0 = 0.5 \log (3\cdot 0.6)+\log \tilde{W}_0, \end{align*} and solving this, we get \begin{align*} \tilde{W}_0 &= \sqrt{\frac{2.625\cdot 0.525}{3\cdot 0.6}}\cdot W_0 \\ &= 0.875 \cdot W_0. \end{align*}

In other words, seeing the odds go from 2-1 to 5-1, is equivalent to experiencing a drop in initial wealth of 12.5%.

Another approach to this (which is described in the linked paper), is to consider what the fair price of bet one is, in the case where bet two has value 0 (can be entered freely).To recap:

- Bet one pays out 2 on a win and -1 on a loss.

- Bet two pays out 5 on a win, and -1 on a loss. Now lets make a combination of bet one and two, so that is worth 1 in both outcomes, i.e. \begin{align*} -w_1 - w_2 &= 1 \\ 2w_1 + 5w_2 &= 1 \end{align*} Solving this, we get $w_1 = -2, \ w_2 = 1$.

Now, since holding $(w_1, w_2)$ units of bet one and two respectively, must be worth exactly 1, and since bet two has value 0, we can find the fair value of 1 unit of bet 1: $$ v w_1 = 1, $$ and hence $v = -0.5$