When Should We Expect Value?

Exploring the Expected Value Paradox

A version of this article appears in my book, Twisted Logic: Puzzles, Paradoxes, and Big Questions (Chapman and Hall/CRC Press, 2024).

UNDERSTANDING THE EXPECTED VALUE PARADOX

At its core, the Expected Value (EV) Paradox invites us to examine how outcomes deviate when we analyse them through the lens of a single ensemble event (a large group participating in an event once) vs. a multiple time event (a single individual participating in the event multiple times).

Take the example of a hypothetical coin-tossing game where players gain 50% of their bet if the coin lands on Heads and lose 40% if it lands on Tails. This game seems favourable for the player—the game has what is termed a positive expected value.

However, the paradox arises when the concept of time is introduced into the equation. While the game appears favourable in theory, it could lead to a net loss for an individual playing this game multiple times. As the coin is tossed more and more, the individual’s wealth may diminish over time, leading to a scenario where they lose all their money, even though the theoretical gain from playing the game is positive.

THE EXPERIMENT

Let’s set up an experiment involving a coin-tossing game with 100 participants, each with an initial stake of £10, to illustrate the difference. In this scenario, we’re employing what’s known as an ensemble perspective, where we’re examining a large group participating in an event once.

Statistically, given a fair coin, we would expect roughly half of the coin tosses to land on Heads and half on Tails. Therefore, of the 100 people, we predict that around 50 people will toss Heads and 50 will toss Tails.

If the coin lands on Heads, each of the 50 players stands to gain 50% of their stake, which is £5. In total, this translates to a combined gain of £250 (50 players × £5).

On the other side, if the coin lands on Tails, each of the remaining 50 players loses 40% of their stake, which is £4. This accumulates to a total loss of £200 (50 players × £4).

Subtracting the total loss from the total gain (£250 – £200), we find a net gain of £50 over all 100 players. When we average this out over the number of players, we see an average net gain of £0.5 (50 pence) per player (£50 ÷ 100 players), or 5% of the £10 initial stake.

THE PARADOX

The Expected Value Paradox becomes evident when we shift from an ensemble perspective, involving many people playing the game once, to a time perspective, involving one person playing the game multiple times.

Let’s examine a scenario where a single player engages in four rounds of the game, starting with a stake of £10. For simplicity’s sake, we’ll assume an equal chance of landing Heads or Tails—therefore expecting two Heads and two Tails.

When the coin lands on Heads in the first round, the player gains 50% of their stake, increasing their wealth to £15 (£10 + 50% of £10). If the coin lands on Heads again in the second round, their wealth grows to £22.50 (£15 + 50% of £15).

However, the game changes when the coin lands on Tails in the third round. The player loses 40% of their current wealth, reducing it to £13.50 (£22.50 minus 40% of £22.50). If the coin lands on Tails again in the fourth round, the player’s wealth decreases further to £8.10 (£13.50 − 40% of £13.50).

Despite starting the game with a positive expected value, the player ends up with less money than they started with. Even though the probabilities haven’t changed, the effects of winning and losing aren’t symmetric.

Thus, the Expected Value Paradox is clear in this example. When many people play the game once (ensemble averaging), the average return is positive, aligning with the expected value. However, when a single person here plays the game multiple times (time averaging), the player loses money.

TIME AVERAGING AND ENSEMBLE AVERAGING

In understanding the Expected Value Paradox, we are introduced to two different types of averaging: ‘time averaging’ and ‘ensemble averaging’.

TIME AVERAGING

‘Time averaging’ is a concept that comes into play when we are observing a single entity or process over an extended period. In the context of our coin-tossing game, time averaging refers to tracking the wealth of a single player as they participate in multiple rounds of the game. Over time, this player’s wealth fluctuates, often resulting in an overall loss despite the odds being in their favour. A severe loss (like bankruptcy) at any point can end the game for the player.

In our coin-tossing game, this would be akin to observing 100 players tossing the coin once. The overall gain camouflages the individual experiences, which can significantly vary—some players win, some lose.

ENSEMBLE AVERAGING

The ensemble average gives us a snapshot of the behaviour of many at a specific moment in time. The ‘ensemble probability’ refers to a large group’s collective experiences over a fixed period.

TIME VS. ENSEMBLE AVERAGING

This difference between ‘time probability’ and ‘ensemble probability’ underscores that a group’s average experience does not accurately predict an individual’s experience over time.

Understanding the distinction between these two types of averaging is crucial when interpreting outcomes of games, experiments, or any process involving randomness and repetition over time. This differentiation becomes especially important in fields like economics and finance, where these principles can guide strategy and risk management.

Strategies that work on an ensemble basis may not be effective (or could be disastrous) when applied over time by an individual—a paradox manifested clearly in our coin-tossing game.

SURVIVORSHIP AND WEALTH TRANSFER

Survivorship and wealth transfer are key elements in understanding how wealth moves around in situations like gambling and investing. The term ‘survivors’ refers to those who keep playing the game through various rounds, while ‘non-survivors’ are the ones who quit, or are pushed out, often because they’ve lost most or all of their money.

The idea is that the wealth lost by non-survivors doesn’t disappear. Instead, it gets transferred to the survivors, redistributing wealth within the system. Take a coin-tossing game as an example: if half of the 100 players lose everything and leave, while the other half double their initial amount, the group seems to break even. But, half of the players have nothing, while the other half have doubled their money.

CONCLUSION: THE INDIVIDUAL AND THE GROUP

In the conventional, or ensemble, view of probability, we look at the outcomes of many trials of an event and calculate averages. Some will win, some will lose, but overall the average outcome should reflect the true odds of the game. The individual variations or ‘paths’ of each person aren’t considered—we’re only interested in the average outcome. This so-called ensemble perspective is often used in classical statistics and probability theory. In contrast, the path-dependent view recognises that the order of events matters.

Take a person who plays a game 100 times. Even if the odds of each game are in their favour, they could still lose all their money if they have a run of bad luck. In this case, looking at the overall or ensemble average wouldn’t accurately reflect the individual’s experience.

In summary, while the ensemble view can provide a broad understanding of expected outcomes, the path-dependent view provides a more nuanced understanding of individual experiences.