MOTA

Mean-Optimized Tournament Allocation: Extending Kelly Principles to Selling Action in Poker Tournaments

As a professional tournament poker player, the key to sustainable success lies not just in having a positive expected value (EV) but in optimizing the growth of your bankroll over countless events. The Kelly criterion offers a theoretically sound framework for determining the fraction of your bankroll to risk to maximize long-term growth. But how do we handle more complex, real-world scenarios, such as selling action at a markup?

From Classic Kelly to Mean-Optimized Allocation

The classic Kelly criterion states that if you know the probability distribution of outcomes and their payoffs, you can determine the fraction of your bankroll to invest to maximize the expected logarithmic growth of that bankroll. If your expected value (EV) of a bet is greater than 1 (profitable), Kelly gives you a fraction to invest. If EV < 1, the Kelly optimal fraction is zero, as you’re facing a losing proposition in expectation.

In a poker tournament setting, calculating a single “Kelly fraction” starts with modeling your probability distribution of outcomes. For example:

• 80%-90% bust with zero return.
• Various ITM (in-the-money) finishes at increasing payout multiples of the buy-in.

By scaling these payouts to match your estimated ROI, you can derive an EV and run a Kelly calculation to find your optimal fraction of bankroll to invest if you were to take 100% of your own action. This fraction ensures you’re growing at the fastest logarithmic rate if you repeat tournaments with the same parameters infinitely many times.

Incorporating Action Selling as a Second Bet

Now, introduce the concept of selling action at a markup as effectively adding another “layer” to your bet decision. Consider two linked bets:

1. The Tournament Bet: You pay a buy-in and receive a payout according to the probability distribution. On average, if your ROI is positive, this is a profitable proposition but with significant variance.
2. The Action-Selling Bet: When you sell a fraction of your action at markup, you’re essentially selling a portion of your tournament equity at a premium. This is akin to taking a side bet: your backers pay more than the face value of the piece of the buy-in they receive. If the markup is sufficiently high, this “sale bet” can carry a positive EV in itself. Even if the tournament alone might not warrant a large Kelly fraction due to variance, the combined proposition of buying in and selling action can alter the effective EV and volatility profile of your overall investment.

By treating selling action at markup as a second bet layered atop the primary tournament bet, you can incorporate it into the Kelly framework. Now the effective “investment” is not just your own buy-in fraction but also the guaranteed inflow from selling. If you’re selling at a markup that exceeds your expected ROI, you’re improving your risk-reward profile and potentially allowing for a higher Kelly fraction than you would take on if playing 100% of your action.

Mean-Optimized Tournament Allocation

This approach, where you calculate the Kelly fraction both with and without selling action, and then choose the configuration that yields the highest long-term bankroll growth, can be described as “mean-optimized tournament allocation.” Essentially, you’re optimizing how much of the tournament you take on yourself and how much you offload to backers at a profitable markup. The steps might look like this:

1. Base Calculation (No Selling): Start by ignoring action selling and calculate the Kelly-optimal fraction. For example, with your given ROI and variance structure, the Kelly fraction might suggest investing 3% of your bankroll. This gives you a baseline expected growth rate.
2. Add Selling Action at Markup: Suppose you can sell 50% of your action at a 1.2 markup (20% above break-even). This changes your net cost and risk profile. Your downside is reduced because the money from selling action partially or fully covers the buy-in. Now run the Kelly calculation again on the “combined bet”:

• Your net outlay might be reduced.
• Your upside distribution changes because you keep a smaller fraction of the biggest scores, but you started off with a profit from the sale. Due to these dynamics, the Kelly fraction for the combined scenario could differ significantly. You might find that by selling 50% at 1.2 markup, you can afford to commit a larger fraction of your bankroll to this event (and future similar events) and still maintain or increase your long-term growth rate.

1. Compare Scenarios and Choose the Optimal Allocation: After running both scenarios, you compare:

• Playing fully on your own.
• Selling some (or even all) of your action at markup and adjusting your fraction of bankroll invested accordingly. If selling action with markup leads to a higher expected log growth rate, you choose that approach. In some cases, if the market for action buyers is extremely favorable (imagine you can sell at a markup that far outstrips your actual ROI), it could even make sense to sell so much that you essentially become an “action broker” rather than a self-funded player. Conversely, if you cannot secure a profitable markup, playing on your own might yield higher growth.

Properties and Advantages

• Extending Kelly’s Properties: The Kelly approach’s valuable properties—such as bounding ruin probabilities and ensuring long-term growth—still apply when you add a second bet (selling action). You’re still maximizing expected log growth, but now over a combined proposition: the original tournament investment plus the immediate gains from selling action.
• Potentially Outperforming “Pure Kelly”: Classic Kelly assumes a single investment. By introducing action-selling, you’re effectively reshaping the payoff distribution, potentially leading to a scenario that is strictly better than pure Kelly in terms of growth rate. This can happen if the markup sale consistently offsets your variance or if it effectively increases your EV beyond what the “pure” tournament bet provides.
• Practical Outcome for Poker Players: In practice, a poker player can use these calculations before each tournament to decide:
  • How much of their action to keep?
  • How much to sell?
  • At what markup to offer it?
  • How large a fraction of their bankroll to allocate given the modified risk-reward profile? Repeating this process for every event leads to a dynamic, data-driven staking strategy. As your ROI estimates change or as the markup market evolves, you adapt your fractions to continuously maximize long-term bankroll growth.

Conclusion

Mean-optimized tournament allocation is a natural extension of the Kelly criterion into the realm of selling action at markup. By modeling your tournament buy-in and markup sale as two interconnected bets, you can apply the same information-theoretic, growth-oriented logic that Kelly betting provides. The result is a more flexible and potentially more profitable staking strategy. When done correctly, selling action can enhance your bankroll’s growth rate beyond what pure Kelly would suggest, allowing you to exploit market conditions to achieve the optimal long-term growth of your poker career. I hope whoever reads this enjoys it. As far as I am aware, this is my original work.