Take the Money and Run: Unpacking YouGov's $50,000 Dilemma

Would you rather have a guaranteed $50,000 or flip a coin for $1,000,000? That was the question asked by the pollster YouGov to some 16,000 Americans and 4,500 Britons at the end of July [1, 2]. Perhaps surprisingly, 65% of the Americans chose the guaranteed $50,000. In Britain, the distribution was even more lopsided. 73% of Britons were unwilling to gamble for a million pounds, against only 21%who were prepared to do so.

Financial commentators pounced. After all, the result makes little sense using conventional frameworks. The expected value of taking the guaranteed money is of course just $50,000. By contrast, the expected value of the coin flip is $500,000 (that is, 50%of $1,000,000): ten times as much. It looks very much like a textbook error.

The reasoning that has led many observers to smugly claim that the average person doesn't understand simple mathematics follows the so-called expected value theory. The question of interest is quite simply: what is the average return, when weighted by outcome probabilities? It is commonly used as a rough shorthand for expected value in low-stakes, everyday decisions, where wealth effects are negligible and the two frameworks give nearly identical answers. For a one-off, life-changing sum, however, expected utility theory is the more theoretically rigorous of the two, precisely because it accounts for something expected value ignores entirely: that the same dollar amount can carry a different weight depending on who receives it and what state of the world they are already in.

In the appendix, I prove that it can be mathematically optimal to take the guaranteed sum of money. However, the reasoning can be understood intuitively. People have concave utility in money - the more they have, the more they need to gain to be equally satisfied with their gains. $50,000 is a trifling sum to Elon Musk, but for many survey respondents it may represent the ability to pay off medical debt (this question was asked in the U.S., after all) or put down a deposit on a new house. The next $950,000 is simply not valuable enough to outweigh the certainty of the first $50,000.

This has applications beyond personal consumption decisions. Banks are increasingly having to take account of the same trade-off when constructing portfolios, as concentration risk in markets has grown alongside the AI investment boom. In a note published on 29 July 2026, Goldman Sachs Research's head of Asset Allocation, Christian Mueller-Glissmann, warned that the average investor's portfolio has become more exposed to the specific companies driving the AI boom (some might say bubble) [3]. His prescription is not to chase a higher expected return, as expected value alone might suggest - it is to sacrifice some of it to reduce volatility. Goldman Sachs Research recommends diversifying into real assets such as infrastructure, property, energy, and gold; low-volatility and high-dividend equities; non-US markets; and options contracts and alternative assets that are generally reasonably resilient during tech-led selloffs [3]. None of this is forecast to raise the portfolio's expected return above what an all-in bet on AI might deliver. The entire purpose is to reduce the variance around that return - which, as the appendix shows, is exactly the operation that raises an investor's certainty equivalent toward their expected value, even while leaving the expected value itself unchanged.

The ongoing Iran war has made this problem even thornier. Since the Strait of Hormuz was blockaded in March 2026, oil and gas markets have been severely disrupted with no immediate end in sight, sending oil prices sharply higher - feeding straight into inflation, with euro-area inflation jumping to 2.5%in March, above the ECB's target [5].

What makes this particularly awkward is that the usual hedge has stopped working. Government bonds are supposed to rally when stocks fall, the classic 60/40 strategy. Instead, global bond yields have risen sharply even as equities have mostly recovered their initial losses, because bonds are pricing in the same inflation shock unsettling everything else [6]. As Bloomberg noted, the oil spike pushed Treasury yields higher instead of lower, breaking the standard playbook of bonds as a safe haven [4].

This is a correlation problem, not just a variance one. Diversification only lifts the certainty equivalent if the assets don't move together; when stocks and bonds sell off together, a "balanced" portfolio's effective variance (and its certainty equivalent) is far worse than the asset mix would suggest. Consequently, many investors have started leaning heavily on gold and infrastructure precisely because they have stayed uncorrelated with both equities and rates, at least for now. The coin-flip logic now applies to portfolios whose traditional hedges have become increasingly unreliable.

The coin flip question visited on hapless survey respondents, portfolio diversification and the bond market that no longer behaves as it should are all kind of the same problem. None of them are really asking "what's the expected payoff?" They're much more interested in the question "what can I actually stomach losing?" - and increasingly, in markets as much as in that viral poll, the honest answer is: less than might be expected.

Consider a consumer with current wealth $w_0$. As in the problem, the consumer faces the option set:

Guaranteed sum: receive $g$ for certain, ending wealth $w_0 + g$.

Take the gamble: with probability $\frac{1}{2}$, receive $J$, ending wealth $w_0 + J$; with probability $\frac{1}{2}$, receive 0, ending wealth $w_0$.

Consider a simple concave utility function. This one is chosen because it is tractable, but alternatives are available:

It is increasing ($u'(w) = 1/w^2 > 0$) and concave ($u''(w) = -2/w^3 < 0$), which is all we need: concavity means each extra dollar is worth less as $w$ grows, so a sure gain is preferred to a fair gamble of equal expected value. This aligns with the economic principle of diminishing marginal utility: the 100,000th dollar is less highly valued than the 1st.

The certainty equivalent $C$ is the guaranteed amount that gives the same utility as the gamble:

Substitute $u(w) = -1/w$:

Because the probability of winning the gamble is $\frac{1}{2}$, $w_0 + C$ is the harmonic mean of $w_0$ and $w_0 + J$:

Subtracting $w_0$ and simplifying gives a clean closed form:

The guaranteed sum beats the gamble exactly when $g > C$, i.e.

Solving this inequality for $w_0$ gives a wealth threshold $w_0^*$ below which the guaranteed amount is optimal:

Substituting the numbers from the problem, we find that:

So anyone with liquid wealth below about $55,556 has $C < \$50,000$, and the guaranteed sum is the utility-maximizing choice. For example:

both below $50,000 - confirming the guaranteed sum wins for typical liquid-savings levels, while $C(w_0 = \$100{,}000) \approx \$83{,}333$, which exceeds $50,000, so the gamble wins once wealth clears the threshold.

Depending on interpretation (if the question is treated as being in total isolation from other wealth), it may follow that $w_0 = 0$ and hence taking the guaranteed sum is always optimal.

N.B. This piece of mathematics is designed for illustrative purposes and does not constitute a formal proof of the optimality of taking the guaranteed sum. In particular, the result is not agnostic to the choice of utility function. The Constant Relative Risk Aversion family of utility functions is given by:

where $\gamma$ is the coefficient of relative risk aversion. Larger $\gamma$ means more curvature in $u$, and hence a stronger preference for certainty. The function used in the main text, $u(w) = -1/w$, is the special case $\gamma = 2$. In the special case where $\gamma = 0.5$, for example, the result will yield a utility function of:

In this case, taking the gamble is always optimal.

[1] YouGov (2026). "Which would you pick: instantly receiving $50,000 or flipping a coin for a 50% chance to win $1 million?" Daily Question, 27 July 2026.

[2] YouGov (2026). "Which would you pick: instantly receiving 50,000 or flipping a coin for a 50% chance to win 1 million?" Daily Question, 22 July 2026.

[3] Mueller-Glissmann, C. (2026). "Balancing the Risks to Portfolios from an Innovation Boom and Inflation." Goldman Sachs Research, 29 July 2026.

[4] MacKenzie, M. (2026). "Iran Conflict Is Tearing Up the Bond Market's 2026 Playbook." Bloomberg.

[5] CNBC (2026). "Stocks, Bonds and Commodities: How Global Markets Have Traded the Iran War." 31 March 2026.

[6] CNBC (2026). "Stocks Under Pressure as Correction Fears Grow and Record Rally Defies Geopolitical Turmoil." 20 May 2026.