Positive sum doesn't mean "win-win"

loops
Sun, 05 Apr 2026 02:33:00 GMT

A lot of people and documents online say that positive-sum games are "win-wins", where all of the participants are better off. But this isn't true! If A gets $5 and B gets -$2 that's positive sum (the sum is $3) but it's not a win-win (B lost). Positive sum games can be win-wins, but they aren't necessarily games where everybody benefits. I think people tend to over-generalize from the most common case of a win-win.

E.g. some of the claims you see when reading about positive-sum games online:

A positive-sum game is a "win-win" scenario in game theory and economics where participants collaborate to create new value, ensuring all parties can gain or benefit.

[Win-win games are] also called a positive-sum game as it is the opposite of a zero-sum game. – Wikipedia

Here I use "positive-sum game" to refer to resource games that involve allocating resources, not allocating utility. "Positive-sum game" isn't a meaningful thing when referring to utility because the utility of each participant can be individually rescaled, so you can turn any game into one with an arbitrary sum; the sign of the sum doesn't matter.

There are a lot of cases where we can make the world as a whole better while simultaneously making some people worse off, and it's important to acknowledge that. Here are some positive-sum situations:

A new innovation benefits most people but puts people who worked on a legacy system it replaces out of a job

Several companies race to create a strongly beneficial invention and capture the market, benefitting the winner and the public a lot, while the losing companies end up having wasted resources

Expropriating someone's land without compensation to build train tracks that are used by a lot of other people

One interesting thing about positive-sum games with losers is that the players can sometimes turn it into a win-win for everybody by having the winners distribute a portion of their winnings to the losers. You can turn positive-sum games into win-wins if:

1. Winners gain transferrable resources (without transaction costs)

2. The resources can be divided into arbitrary portions

3. The amount of gains/losses that accrues to each player is known to everyone

4. Players can precommit to transfer resources after the game (otherwise the winners can defect and not transfer the resources)

This is the concept of turning a Kaldor-Hicks improvement (an improvement where everyone would hypothetically be better off if the winners compensated the losers) into a Pareto improvement (an improvement where everyone is better off).

One interesting example is an efficient auction with an entrance cost[1], which benefits the winner (who values the good the most) and auctioneer, and harms all the other bidders (who paid the costs of entering into the auction and got nothing). The entrance cost doesn't need to be a direct fee to enter into the auction; it can also include indirect costs like spending time and effort to decide how much to bid.

The winner's consumer surplus (how much their value of the goods exceeds what they paid) is value to them, but not cash that they can transfer to compensate the losers. If the winner has enough money they could compensate the other bidders for their wasted costs of entering the auction, and everyone would be better off, but if not the auction winner is better off but can't compensate the losers. In practice, valuing the indirect costs bidders have for entering into auctions is difficult and so auctions are often positive-sum games with losers.

Another example interesting example is expropriation, in practice the government usually pays the fair market value of the land to the person whose land was seized, attempting to turn a positive sum game with losers into a win-win, although landowners often feel the expropriation payments aren't sufficient.[2]

I think it's important to keep all this in mind when making positive-sum proposals that there might be losers and they should be compensated if possible; "positive-sum" doesn't mean that everyone benefits.

1. ^

This is only positive-sum if the surplus for the winner exceeds the total entrance costs for all the bidders, which I assume is the case.

2. ^

Which makes sense: landowners have a revealed preference that they value their land more than the fair market value, because if they valued it at less than FMV they could just sell it for the FMV and be better off. (Ignoring illiquidity and the transaction costs for selling the land.)

Discuss