Prospect theory and why you should quit playing Elo based games like overwatch

What is an Elo based ranking system?

The essence of Elo is that it ranks players by points based on games won and lost. After every game, the winning player takes the losing player’s points. The ranking is comparative; it represents how players fare against another player, and it does not have an interpretation on its own.

Of course, we can never truly know how a ranking system works in a game unless the developers tell us how they have designed it. However, generally, most competitive games adopt a similar ranking system to Elo. They match high-ranked players with high-ranked players. If we take overwatch as an example, the practical effect is that all players in a match are at a similar skill level.[1]

An analysis of why an Elo based game might be bad for players

Assumptions

This article will make a few assumptions throughout this analysis. The assumptions are not necessarily true for everyone and are dependent on individual preferences.

1. Players feel good when they gain points in rank. Their utility gain from winning a game is denoted by W.
2. Players feel bad when they lose points in rank. Their utility gain from losing a game is denoted by -2W.
   • Prospect theory suggests that a value function is steeper for losses than for gains. In other words, a loss is more painful than a gain. Tversky and Kahneman have estimated that a small loss is 2.25 times as large as a utility gain of an equal amount.[2] For example, investors are more willing to sell shares that have gained in value and are less willing to sell shares that have lost in value. This is because losses are more painful to realise than gains. For simplicity, the article will use a coefficient of 2 here.
   • One may disagree with this proposition. Adopting this assumption would lead to an absurd conclusion that every game which involves counting wins and losses is not worth playing if the chances of winning are 50/50 and if wins and losses completely determines utility. This is because the expected utility will always be negative since losses are more painful than gains. However, the distinctive feature of an Elo system is that players treat their points like money. Players take their rankings seriously, and they behave as if they are dealing with money. In other words, a loss of points is like a loss of money; a gain in points is like a gain in money.
3. Players feel good when they and their teammates have done a good job. Their utility gain is denoted by G.
4. Players feel bad when they and their teammates perform poorly. Their utility gain is denoted by B.
5. For the simplicity of calculation, this article assumes the chances of 3,4 occurring is 50%.
6. The above completely describes the utility changes of a player in a game.

Analysis

Based on the assumptions above, a player can have four possible combinations of utility gains from playing an Elo-based game.

They are:

1. Win the game + good job = G + W
2. Win the game + bad job = G – 2W
3. Lose the game + good job = B + W
4. Lose the game + bad job = B – 2W

Since the matchmaking system aims to match people with people who are at a similar skill level (same rank), the probability of one player winning is 50%. The expected utility gain from a player who plays other players who are at the same skill level is then: (2G-W)*0.5 + (2B-W)*0.5 = G+B-W

Therefore, a player will consistently lose utility if the size of G and B are the same. In other words, the player assigns the same absolute values on whether they perform good or bad. Using notations, |G| = |B|.

Alternatively, as long as the difference between G does not exceed B by an amount more than W, the player will consistently be worse off.

Discussion

The result suggests that if a player plays at his rank, he will lose utility provided that his valuation of good performance does not exceed bad performance by more than the value of a win in the game.

The result explains why developers reset the rankings in new seasons. This is to encourage those who are playing at their rank to gain utility because their odds of winning are greater when they play at a lower rank. Hence, have a greater utility gain. This also offers some explanation as to why there are players who deliberately create new accounts so they can play at a lower rank than their main account. It is because their chances of winning are larger, which may result in more utility gains.

Why do people keep on playing if they are gaining a negative utility?

Players like to finish feeling like they have won the game and not feel like they have lost the game. This is consistent with the idea of ‘using zero daily profit as a reference point and gambling in the domain of losses to break even’.[3] Players would keep on going and can never stop when they play at their rank since it is impossible to break even because they lose utility even if their rank stays the same. This is because a decrease in rank is much more painful than a gain in rank.

It seems impossible to gain utility except for the first few matches when playing Elo based games. This conclusion is subject to the assumptions and each individual’s valuation of W, G and B.

The real problem is that we take Elo rankings too seriously and treat them almost like something we own. In a classic game that does not use the Elo system, we have less attachment to wins and losses. For that reason, it seems the way to enjoy games is to enjoy the game itself and care less about whether you have lost or won. I hope everyone who keeps playing games can enjoy the game casually and be less attached to the relative ranks of your skills.

[1] Horowitz, D. (2020). What Are Overwatch Ranks? Retrieved from https://www.hp.com/us-en/shop/tech-takes/what-are-overwatch-ranks.

[2] Tversky, A., & Kahneman, D. (1992). Advances in Prospect Theory: Cumulative Representation of Uncertainty. Journal of Risk and Uncertainty, 5(4), 297-323. Retrieved from http://www.jstor.org/stable/41755005.

[3] Camerer, C. (1998). Prospect Theory In The Wild: Evidence From The Field. Advances in Behavioral Economics.