"Zero Sum," Framing Games, and Pareto
Why zero-sum is about context, and positive-sum is different than pareto improving,
In the last post, I quoted Russ Roberts saying that “Competition can be a form of cooperation even when the game looks zero-sum.” But I should explain what zero-sum thinking is, and what it isn’t, and then I’ll talk a bit more about framing the games, and how games get transformed.
What Zero-Sum Is (And Isn’t)
The term has morphed from being a technical description of a certain class of game in game theory to being a much more general concept - losing exactitude along the way. I’ll explain below that I think this makes people use the term in somewhat muddled ways, but first, a definition: in a “zero-sum game” the payoffs to different players of a game have a fixed sum. The simplest case is win-lose games, where there is a single winner, and one or more losers, but no ties. Much come complex payoff structures can also be zero-sum; any game where the sum of the payoffs to different players is some fixed amount is called zero-sum. The term, in fact, comes from the fact that we can write any game with fixed total payout as a game where the sum of all payoffs equals zero; just subtract (or add) the fixed amount. But as we’ll discuss below, the sign of that constant is very important.
Game theory, of course, is a formalization of human interactions in games, and the choice of exactly how to formalize an actual situation is far less exact that the mathematical abstraction. As Leopold Kronecker wrote, "God made the integers; all else is the work of man" - and for that reason, the formalizations we use to describe reality can mislead; not everything that you might formally describe as zero-sum actually needs to be.
As we noted before, as long as the total payoff is fixed, a game is zero-sum - players do not create or destroy wealth in the game. Poker is a zero-sum game - everyone gets out some fraction of what everyone brings to the table, and every dollar they win takes it away from the total available to others. On the other hand, splitting bills among apartment mates is also a zero-sum game. Should they split the cost evenly, or by room size, or based on their different incomes? Many different ways are reasonable to propose, but the total paid is fixed. The difference is that the fixed amount is a cost - there wasn’t any benefit, and no-one “won” the negotiation. Lastly, cake-cutting is another example of a zero-sum game, where a group of people want to split a cake. The entire cake is available no matter what, the question is how to split. But in this case, we find that the discussion in the previous post is relevant, because different preferences lead to cooperative mutually beneficial solutions; my kids want more icing, I want more chocolate cake, and everyone can get more of what they want, even if the total amount of cake stays the same.
And this shows us that my changing the framing - in the previous example, from “how much cake” to “what do you want” - you can make zero-sum games into effectively cooperative positive sum interactions. To see a few more ways this can happen, we can talk about a salary negotiation, where the employer and employee are bargaining. This is cooperative, even if it is zero-sum; both would prefer a deal. At the same time, every dollar paid by the employer is a benefit to the employee at the cost of a dollar to the employer, the very definition of zero-sum. But if that were the entire picture, the employer wouldn’t hire anyone; they would gain nothing! Instead, we should look outside the narrow game to see that both sides will benefit, as is true in any business deal, and in this case the employer expects to make more money because of the employee that the employee is paid. And there are many ways to expand the framing of a “zero-sum” problem to see the hidden positive sum game. But even without expanding the frame to look at the broader benefits, we can reframe the game in many ways to avoid framing the negotiation as zero-sum.
Benefits that aren’t taxed cost the employer less than the employee gains, so the total can be larger than the sum of the benefit1.
Employers can offer benefits like flexible work hours or on-premise amenities that are worth more to employees than they cost.
Employers can offer non-monetary benefits, like status, job titles, access to cool technologies, and interesting or fun jobs that people want to do. And these don’t need to be at the expense of what benefits the company. For example, Google’s 20% policy (even if few employees use it) is a way for Google to get people to build Google products they are interested in building.
And if we do expand the frame slightly, employers could pay a smaller salary plus a bonus. If the bonus is a faction of the employers profits, the employee potentially makes more, but so does the employer. (And we’ll definitely talk more about these principle-agent interactions in a different post!)
Maybe all of this talk about how the world doesn’t need to be zero-sum seems obvious, but the below anecdote, via Abram Demski’s previously cited essay, shows that this isn’t obvious to everyone.
In the introduction to the 1980 edition of The Strategy of Conflict, Thomas Schelling discusses the reception of the book. He recalls that a prominent political theorist “exclaimed how much this book had done for his thinking, and as he talked with enthusiasm I tried to guess which of my sophisticated ideas in which chapters had made so much difference to him. It turned out it wasn't any particular idea in any particular chapter. Until he read this book, he had simply not comprehended that an inherently non-zero-sum conflict could exist.”
(via Abram Demski)
Positive Sum and Negative-Sum
The most famous game in game theory, of course, is the Prisoner’s Dilemma. And it’s definitely not zero-sum - the entire point is that cooperation is better for everyone, but unstable. However, we can pose the prisoner’s dilemma in two different ways that are equivalent as games, but vastly different as modes of interaction. And this gets to a critical issue of why game theorists don’t talk about three types of games, they talk about only two; zero-sum, and non-zero sum. That is, positive sum and negative sum games are the same - but as I’ll argue, they are also completely different.
In the original case, which gave the scenario its name, there are two accomplices who were caught, and can either stay loyal to each other, or work with the police to provide evidence that ensures the other will be punished more severely, with the promise of a slight reduction in their own sentence. Of course, it’s better for the prisoner to cooperate than for both to defect, since they will end up with more severe punishments if the other defects to the police - but the dilemma is that when playing a single time, given what the other player chooses, the prisoner is rewarded for defecting. But critically, the best outcome, when they cooperate, is for both prisoners to end up with light prison sentences. The game is negative sum, in that no matter what happens, they players are worse of than before they were caught.
On the other hand, there is a TV show, Golden Balls, which makes contestants play a monetary version of this game in practice. To start, there are several rounds of play, in which the players can lie or not, building to the final game, where the two final contestants choose either to split the prize, or to steal. If they cooperate, that is, split, then as the name implies, they split the money. If one steals, that player takes the entire total, while if both steal, no-one wins. As a prisoner’s dilemma, each player is better off being the only one who steals, but if everyone is selfish, playing leaves everyone going home empty handed. (Overall, about half split the money.)
Critically, this second game has the same dynamics as the first, but the players can end up rich, rather than just less-severely punished. Yet the two games are effectively identical. And that’s why positive sum and negative sum games are the same, at least when they remain narrow. But we care less about the formalized question than we do about the reality - and the reality of these situations is quite different.
Groups, Cooperation, and Punishment
A variant of the prisoner’s dilemma is the public goods game. In this game, everyone is given some amount of money, and can either put the money in the communal good, or keep it. Every dollar shared gives everyone in the game some fraction of the money spent - say, half. If we have 5 players, and each is given $10, they could all put their money in, and with $50 in the public good, everyone walks away with $25. That’s a great deal, if we can just get everyone to cooperate.
Unfortunately, this is a variant of the prisoner’s dilemma. If everyone else cooperates, and you defect and keep your money, only $40 goes into the communal good, so everyone else gets only $20 - but you walk away with $30. That’s more than you get by cooperating. And everyone faces this choice, so selfishness leads to everyone going home with only $10.
As always, if only people could cooperate, they’d all be better off. But you don’t need to cooperate with just one person. Even if you trust three of the other players, and cooperate, the last player exploits everyone else. But if we expand the game slightly, we can change the interaction completely.
In the new version, we’re playing the same public good game, but there’s another option for each player, in addition to keeping the money, or putting into the public good, they can spend money to buy punishment. In the setup I think is simplest, people can decide how much they want to punish, before they put in the money. Everyone can see how much was committed to punish others, and the punishment happens at the end of the game. If anyone didn’t put in all of their money, for every dollar anyone pays to punish, everyone who paid less than them into the pot loses two and a half dollars.
Now, if everyone except you promises $3 to punish, then puts in $10 and decides to punish, and you put in nothing, you get $30, and they each get $20 - but they decided to punish you, so you lose $2.50*3*4, or $30. You get nothing, and they get to keep $17 each. You’re better off just putting in your money. Is this “really” cooperating, in the sense I discussed several posts ago? Not quite - you didn’t sacrifice anything to benefit others, you just did the selfish thing of promising the punish and putting money into the communal fund, and this happened to benefit others as well. But committing to punish is costly!
In some sense, “cooperating” in this game is paying to punish, getting less money so that others are forced to cooperate, and can’t get away with keeping money to themselves. But we see that by expanding the game options, we can change the dynamics. As the above example shows, in theory, regulation and law enforcement allow cooperation. And in the previous post, I mentioned that “regulation and law enforcement are a key tool that the public uses to fight back [against coercive abuses].”
But the opposite is also true.
Zero-Sum Enforcement and Narrow Coalitions
As the game above showed, there are ways for government to promote positive sum outcomes by enforcing rules that enable positive-sum interactions. Trade and economic growth are, in general, positive-sum, so laws that allow for more trade and better enforcement of contracts are going to be generally positive sum.
Not every way to expand a game is going to be positive. In the previous post, I mentioned homeowners associations as a tool for cooperation - and they can be, when they prevent owners from doing things that ruin the neighborhood for everyone, with no benefit. That preserves value, or increases value, making the deal positive sum. But that’s not the only way they are used. Instead, they are frequently used to create cartels, for instance, restricting homeowners ability to sell their land to developers. This is also preserving value - having an apartment building next to your house makes the house less valuable. But at anyone who has paid attention to the housing market in the past couple decades, this preservation of value has tremendous costs more widely.
That is, rules can make otherwise positive-sum games into zero-sum games. Another example is taxing imports, which is a policy that can promote domestic industry, or be used to ensure that critical supply chains are not fragile to international disruptions - but they are usually pushing for zero-sum punitive measures to protect a narrow industry, preventing far larger public benefits.
That is, cartels and industry protections are ways to cooperate narrowly, but that can prevent greater benefits, defecting from larger society, or not. And even in places where we might say no cooperation is occurring, it’s a question of boundaries. A country might decide to eschew international norms to impose tariffs, but it’s a collection of people that are cooperating. And that’s not always a bad thing! To quote Ben Landau-Taylor, “There are tariffs and then there are tariffs. Tariffs as part of industrial policy to nurture an infant industry, phased out as the industry has a chance to prove itself, usually works great. Tariffs as a random revenue source or barrier to trade is usually a disaster.” That is, if we put the boundaries in the right places, even actions that are otherwise damaging can be broadly beneficial.
But the failure to cooperate at a higher level is usually a lost opportunity, or even a tragedy, happening because of too-narrow cooperation. And this highlights a critical question that will be a theme we’ll return to - the question is not whether to cooperate, but with whom. And it might seem that the wider the set of people cooperating, the better - but as the quote above showed, this isn’t always the case.
To get there, I’ll briefly address a conceptual confusion about what zero-sum doesn’t mean, hopefully treading lightly using a modern political example - and after that, I’ll go back a century and talk about Villifred Pareto.
Zero-Sum Mindset, Positive-Sum Redistribution?
Last week, Eric S. Raymond, famous for his book on the ethos of Open Source, among other things, tweeted about a paper from last year claiming that “younger people in the US and other wealthy nations are far more likely to have a zero-sum mindset—the idea that someone’s gain means a loss for others—compared to older generations.” He tied this to the liberal-conservative divide, blaming communism for the zero-sum mindset, while the original authors asked a related but somewhat more nuanced set of questions. For example, an author said “we ask respondents if they believe gains for some come at others' expense in different relations: between countries, racial groups, immigrants vs non-immigrants, and income groups.”
I think the focus on where gains come from is somewhat confused. Sure, denying that positive sum interactions can take place is a zero-sum mindset. But it certainly can be the case that gains for some come at the expense of others, even in positive sum games. The problem, of course, is that experiences lead to preferences. And exploitation is a historical fact that people are aware of. So the paper, not surprisingly, found that those who had zero-sum mindsets were more likely to support different policies - and not only along political lines. “Zero-sum thinkers” on the right supported immigration restrictions and universal healthcare, while “zero-sum thinkers” on the left supported high taxes on the rich and affirmative action. As the original tweet noted, when people believe that interactions are zero-sum, it makes sense to do different things than if they believe they are positive sum, or could be made to be so.
And as the author noted, “older generations experienced much higher growth and better economic conditions when they were young, which could have reduced their zero-sum world views… historical coercion of one’s ancestors, such as forced reservations, indentured servitude, and concentration camp imprisonments during the Holocaust, are linked to stronger zero-sum views today.” But slower growth and historical tragedies do not imply the games are zero-sum.
It should not surprise anyone that people who have personal or historical experience with exploitation believe that gains for some can or do come at the expense of others - it’s a rational conclusion given their experience. And of course, when there is no net economic growth then any (purely economic) interactions are, in fact, zero sum. But even if the interactions are positive-sum, that doesn’t imply they are pareto-improving.
Pareto Assumptions And Policies
Vilfredo Pareto was an Italian engineer, sociologist, and economist born in 1848, who came to prominence not only for his mathematical contributions to economics but also for his observation about statistical distributions as it applied to wealth. Trained as an engineer, Pareto noticed that in the world around him, before World War I, wealth and resources followed uneven but predictable patterns across societies. He famously noted that in Italy, about 80% of the land was owned by 20% of the population—a distribution now known as the “Pareto principle” or the “80/20 rule,” and a pattern that seemed to hold across contexts, from land ownership to business profits.
This early observation influenced Pareto’s later work on welfare economics, where he sought a way to evaluate economic improvements that didn’t rely on subjective judgments of fairness or equity. He developed what we now call Pareto efficiency: an outcome is Pareto-efficient if no one can be made better off without making someone else worse off. The notion appealed to Pareto’s view that economics was primarily a mathematical endeavor, by creating an objective criterion for evaluating economic changes. This fit nicely with the ideas his mentor and predecessor, Leon Walras, had about how markets functioned, maximizing value for consumers by negotiated prices - and Kenneth Arrow and Gérard Debreu formalized this basic part of mathematical economics in the 1950s as “Walrasian Equilibrium,” where market will find pareto-efficient solutions via Walras’ ideas about iterated bids finding a price at which trades occur.
Walras followed the Fabians in calling for the nationalization of land (via gradual democratic reform, not communist revolution.) But Pareto came to economics in part due to his frustration with government bureaucracy, and his view was much more libertarian. His version of efficiency allows optimality in a specific sense, but it also enforces any unfair distributions of resources within society. That is, an outcome which is Pareto-efficient leaves inequalities at least partly intact, because any attempt to improve someone’s position would mean diminishing someone else’s. And perhaps this informed, or was informed by, Pareto deep opposition to the growing trend of socialism.
In today’s policy debates, Pareto efficiency is often invoked to support policies or market structures that prioritize overall gains. It’s seen as a benchmark for maximizing societal benefits without imposing losses. But it does not satisfy people’s sense of fairness, especially in societies where historical inequities and systemic exploitation are deeply ingrained in memory. The zero-sum mindset common among those who’ve experienced or learned about exploitation suggests that people’s tolerance for inequality often depends on whether they feel a system benefits them—or whether gains come at others' expense.
When people see economic benefits going disproportionately to a small segment of society, they’re less likely to view interactions as positive-sum, even if they technically are. And the truth is that inequality has grown, and periods of high and growing inequality are not conducive to
In 1913, a few short years after Pareto made his famous observation, immediately before World War I, Hellen Keller said, of the United States, “the country is governed for the richest, for the corporations, the bankers, the land speculators, and for the exploiters of labor.” To the extent that this is true, Pareto-improvements will reinforce the conditions pushing for what was called zero-sum thinking. On both sides of the political spectrum, politicians favor policies that are not pareto-improving, whether higher taxes on the wealthy, or restrictions on immigration. These attitudes reflect a common belief that fairness requires intervention, even if it means diminishing efficiency in the process.
Non-Pareto “Improvements” and Poverty
Cooperation can exclude some people from benefits and still create positive-sum outcomes. In fact, it’s nearly certain that redistributive policies are positive sum in the short term. As we noted in a previous post, people’s desire for money is roughly logarithmic. In cases where wealth is distributed in the way Pareto observed, it would be shocking if taking some proportion of the wealth from the wealthiest people and distributing it to the poor wasn’t welfare improving, in the short term.
As a simple example, an island with $1000 divided among 5 people in a Pareto fashion would have 1 person with $800, a second with $145, then $30, $10, and $5. Using a simple log utility, the total is 8.29 “points” of utility. On the other hand, transferring $40 from the richest person to give each of the others $10 increases that total to 9.16 - with a cost to the richest person of 0.02 points.
This is a substantial improvement, at minimal cost. But, aside from the fact that this isn’t a pareto-improvement, it reduces incentives for wealth generating activities - and there are strong argument that this isn’t a good decision in the long run. The question here is about how governments balance different goals, and this is a very different set of questions about cooperation, one which we’ll deal with in other posts. However, before we finish, there’s a tremendously important point to make about redistribution; the world is still too poor for it to work.
If total global production was divided evenly among people, each person would get around $10,000 per year - a relative fortune for many, but not enough to support a middle class lifestyle for the world (and this is aside from the fact that the economy would collapse as no-one had any reason to work, and that we do not have a way to implement such a policy, and that much of the wealth cannot be transformed into cash to distribute, and many other reasons this is impossible.) It is clear that today, the world cannot afford fully automated luxury communism, and that will remain true unless and until we have some way of making everyone more wealthy. That’s not impossible, with technologies like AI and automated farming and manufacturing, but it certainly requires cooperation at many levels.
Next up in exploring cooperation
To recap briefly, we explored the concept of zero-sum thinking, from game theory to applications in economic and social interactions. By examining examples like salary negotiations, public goods games, and shifting regulatory boundaries, we saw how reframing narrow, zero-sum conflicts can unlock positive-sum outcomes, while also recognizing some partial limits to cooperation when groups prioritize self-interest over wider benefits. We then connected these principles to historical and contemporary perspectives, from Pareto’s observations on inequality to current zero-sum mindsets in politics, illustrating how beliefs about fairness and redistribution shape policy. Together, this leads to a fundamental question for cooperation—not just whether to cooperate, but with whom, and under what conditions.
So the next post, and possibly the next several posts, will be about the challenges of multi-level cooperation. I’ll start from global geopolitics, and talk about why the question of who to cooperate with leads to complex multi-level games, and (sometimes) makes simple solutions fail. And this will hopefully illustrate why the challenges at the largest and most difficult levels are the reason for (relative) success at lower levels, and point to some reasons for both optimism and pessimism about the future, and about AI.
It might look like this is just robbing Peter (the government) to pay Paul (the employer and employee,) but in fact, tax-deferred retirement savings, insurance, and many other benefits that are untaxed aren’t a transfer from government, they are actually net-beneficial more widely. But that is a broader argument, and as other examples show, not necessary.