Coalitions are everywhere
Introduction to the psychology and game theory of coalitions
Benjamin Franklin is credited with the following comment on the importance of being part of a cohesive group, during the signing of the United States Declaration of Independence in 1776:
We must indeed all hang together, or, most assuredly, we shall all hang separately.
Facing the might of the British army, the colonies needed unity to avoid their rebellion being crushed.
This statement1 highlights a fundamental aspect of human life: the importance of forming alliances for success. Such groups—“coalitions”—are everywhere in our lives. Thriving in society depends on navigating the complex landscape of coalitional games, where there are many players, others’ intentions are uncertain, and possible moves are too numerous to grasp fully.
Our psychology is attuned to these dynamics. We care about which groups exist, where we belong, and what our place is within them. This post introduces the psychology and game theory of coalitions—two perspectives that, despite their complementarity, are rarely integrated.
The foundations of coalitional psychology
There are two extreme views on human social nature. One assumes people can be endlessly altruistic; the other sees them as entirely self-interested.
Both are mistaken. The first underlies ideologies that purport to subordinate individuals to the greater good of the community (e.g., communism, nationalism, and religion). As E.O. Wilson quipped about communism: It is a great theory, but for the wrong species. It better describes ants, which are willing to sacrifice for their colony due to their close genetic relatedness.
The second perspective views society as merely a collection of individuals, an idea often linked to libertarian thought. Margaret Thatcher famously stated:
There is no such thing as society. There are individual men and women, and there are families. - Thatcher (Woman’s Own magazine in 1987)
This view is also inaccurate. Humans do not operate solely as individuals or nuclear families. Unlike solitary birds like robins, we form large, enduring social groups with shared interests.
Humans fall between ants and robins on a social behaviour spectrum. We are neither fully absorbed into a grand collective nor purely self-centred. Instead, we navigate a middle ground—far more cognitively demanding than either extreme. Robins need no coalition strategies, and ants instinctively align on common goals. In contrast, humans must strategise around coalition formation, maintenance, and renegotiation.
Small coalitions of two or three individuals appear in other primates, such as chimpanzees and gorillas (Harcourt and de Waal, 1992). However, large-scale, long-term collective action is uniquely human.
There are several theories about why human ancestors evolved the ability to form larger coalitions. In The Social Leap, Bill von Hippel discusses one such theory: the need for defence against predators as early hominins adapted to a changing environment. Due to climatic shifts, dense forests in parts of East Africa gradually gave way to open savannah, leaving our ancestors more exposed to predators like lions. One proposed defensive strategy that may have emerged in response was the coordinated throwing of stones to deter attacks.
A lone Australopithecus afarensis throwing stones (perhaps while other members of its group ran away) would have ended up in the belly of a slightly bruised predator, but many Australopithecines throwing stones could probably have driven off hyenas, saber-toothed tigers, and even lions. It was this need for collective action that brought about the most important psychological change that enabled us to thrive, rather than just survive, on the savannah: the capacity and desire to work together. - von Hippel (2018)
Dunbar’s Social Brain Hypothesis (1998) proposes that the evolution of human intelligence was driven by the cognitive demands of managing increasingly complex social networks. From childhood friendships to office politics and national governance, navigating social relationships and maintaining coalitions remain central to our lives.
Yet the greatest threat to humans has never been predators—it is other humans. Most violent deaths throughout history were caused by fellow humans.2 Archaeological evidence suggests that, over evolutionary time, many within-group killings involved several individuals turning against one (Boehm 1999). Being isolated is perilous; it leaves one vulnerable to organised aggression. In contrast, a strong coalition offers protection.3 This evolutionary pressure honed our ability to play coalitional games—understanding alliances, shifting loyalties, and strategic positioning (Tooby and Cosmides, 2010).
The features of coalitional psychology
The need to belong
Given the importance of coalitions for success, our psychology is finely tuned to help us navigate complex social relations. A key element is the strong desire to belong and the anxiety that comes with social exclusion. A classic psychological experiment illustrated this: in a simple video game where participants tossed a ball to each other, players became distressed when others stopped including them.4
Navigating coalitional choices
In many situations, people have natural coalitions—family, colleagues, compatriots—but choices still arise. Within families, organisations, and nations, disputes force individuals to take sides. Additionally, modern social mobility offers opportunities to join new groups, potentially at the cost of leaving others.
The challenges of the formation and management of friendship groups show the complexity of the psychology required to navigate coalitional choices. Success requires recursive mind reading: not just assessing potential allies (“Would they be good friends?”) but also understanding their perspectives (“Do they think I’d be a good friend?”) and even anticipating others’ views of their opinions (“Do they think their friends will like me?”). Our reputation and that of others play a central role in shaping these perceptions.
Our deep concern for our reputation
Much of our social behaviour is shaped by concerns about reputation. Our ability to gain cooperation, trust, and resources depends on being seen as competent and reliable.
Economists have traditionally studied this in the context of partnerships—choosing friends, business associates, or spouses. These are minimal coalitions composed of two individuals. The logic can be extended to the formation of larger groups, such as workplace teams or social circles. Being accepted into a coalition often hinges on our reputation.
As a result, much of what we do is aimed at shaping how others perceive us. We care about signalling competence and trustworthiness. Signalling theory, a branch of game theory, examines how actions convey such qualities. For instance, passing difficult exams signals competence, while repaying debts on time signals reliability.
In addition, much of our social communication is designed to provide a positive interpretation of our past actions. It is not just the cases of obvious showing off, it is the topics we choose to discuss, how we frame our past actions, and how we selectively omit details. Often, we are not consciously Machiavellian but engage in self-deception, genuinely believing the narratives we construct. Like lawyers defending a case, we instinctively craft the best possible version of our story.
The importance of influencing others’ perceptions—and accurately assessing theirs—explains why gossip takes up so much place in our social interactions. Psychologist Robin Dunbar estimated that about two-thirds of human discussions are about social gossip, whereby we learn what others have done, why, and how (Dunbar, 2004). Gossip serves not only to share information about other people but also to verify others’ claims and assess trustworthiness.
When Aristotle called humans “political animals,” he was profoundly right.5 This goes beyond interacting on public forums, like the Agora or Twitter. The politics of coalition-building permeate every aspect of our lives, from friendships to workplaces.
The game theory of coalitions
As I seemingly used the term “coalition” for any social group, you might be wondering what is a coalition. The game theory of coalitions has given us ways to think carefully about this question.
What is a coalition?
The foundation of coalitional game theory was laid by von Neumann and Morgenstern in their 1944 book Theory of Games and Economic Behavior. They defined a coalitional game as any situation where you have a group of individuals, who can form coalitions (groups), with each individual belonging to at most one group. Each coalition has a total payoff that depends on the identity of its members.
It is a very general conceptual framework, applicable to a wide range of situations, from friendship groups to high politics.
The gains from being in a coalition
Why do people form coalitions? Because individuals can gain more from being in a group than from acting alone. Key benefits include:
Defence: Strength in numbers, from early humans fending off predators to modern military alliances.
Insurance: Groups offer support when unforeseen setbacks affect some of its members.
Division of Labour: Specialisation within a group increases productivity.
Collective Action: Some things (e.g. moving heavy furniture) can only be achieved with a coordinated effort.
The gains from cooperation are therefore at the heart of coalition formation. Von Neumann and Morgenstern adopted as a fundamental principle of a coalitional game that the payoff of a coalition tends to be larger than the sum of payoffs of its individual members if they were alone (as well as the sum of payoffs of smaller coalitions).
Who should get what in a coalition?
Once a coalition forms, a key question arises: how are the rewards shared? Sometimes the answer is straightforward—if Australopithecus afarensis team up, nobody gets eaten. Other times, the division does not have a unique answer—when Mark Antony and Octavian form an alliance to defeat Brutus and Cassius for control of Rome, how should they agree to split the spoils of victory? [Narrator: They did not reach a peaceful agreement.]
Coalitional game theory does not dictate an outcome when many divisions are possible. However, Lloyd Shapley’s Nobel-winning work (1953) identified three fundamental principles that lead to a clear division:6
Efficiency: The full coalition payoff is distributed—nothing is left over.
Symmetry: Players with identical contributions receive identical shares.
Null Player: A player who adds no value to any coalition gets nothing.
When these principles hold, there is only one way to divide payoffs. It is known as the Shapley value, which is a number that represents a player’s average contribution across all possible coalitions. The intuition is easy to understand: whenever a player joins a coalition, they increase its total payoff. The Shapley value calculates the average of these contributions across all possible sequences of coalition formation.7
Fairness and bargaining in coalitions
Why is the Shapley value a good benchmark to think about how people could split the payoffs from a coalition? The idea is to start from principles that players would find reasonable to agree on, a fair way to split the gains from participation in a coalition. It is hard in particular to reject the principles of efficiency and symmetry.8
We observe that, as a result of these principles, the Shapley value varies based on an individual’s ability to contribute—what somebody brings to the party. If an individual can contribute significantly to any group, their participation in a coalition will be valuable and therefore valued. Others should be willing to allocate a larger share of the pie to this person. If, for instance, you get Lionel Messi on your football team, you accept that he earns a much higher salary because he increases the team’s chances of success.
In addition, the Shapley value also varies based on an individual’s exit options—what somebody could get in another party.9 If you are highly productive but only within a specific subgroup, your exit options are limited, and you may struggle to negotiate a large slice of the pie in that specific group. If, on the other hand, you are in high demand and many groups value your contribution, you will be able to secure a better deal. Indeed, many people find that the best way to get a raise in the professional world is not just by being productive and loyal but by receiving external offers, leading their employer to make counteroffers to retain them.
Why do contribution and exit options determine what someone gets from the coalition? One way to think about it is that if coalitions formed randomly and repeatedly, a player’s expected payoff would reflect what they would receive on average. This fair benchmark can, therefore, also be thought of as reflecting the bargaining power of each player—if coalitions could be renegotiated all the time and each coalition had an equal probability of occurrence, the Shapley value would capture each player’s expected payoffs.10 Indeed, in the context of voting games, where players aim to form majority coalitions, it is referred to as a power index.11
Besides the numbers, this framework helps us think about the key strategic variables likely to influence what you get in a group: your contribution and your exit options. Having either in isolation is not enough. It is having both that makes you essential and likely to get a sizeable share of the coalition payoffs (be it money or social recognition).
In summary, humans are naturally group-oriented. They form coalitions, care about their group membership, strive to be central within their groups, and worry about being marginalised—or worse, excluded altogether. Navigating the shifting landscape of coalition formation and evolution is complex. It requires anticipating how others might form coalitions, which groups would benefit or harm us, and what coalitional move would yield the best outcomes. This may involve strengthening an existing coalition, modifying its composition, defecting to another, or preventing the formation of a rival group. From groups of friends, and office colleagues, to the cut-throat environment of politics and power, coalitional dynamics are everywhere and they explain much of what we do and think of daily.
This is the first post in a series on coalitional psychology and game theory. In future posts, I will use these insights to discuss a wide range of social phenomena, from politics to everyday life. The next post will apply the game theory of coalition formation to explain recent shifts in political coalitions in Western democracies.
References
Aristotle (c. 350 BCE). Politics. Edited by W.D. Ross (1908). Oxford: Clarendon Press.
Aumann, R. J., & Drèze, J. H. (1974). “Cooperative games with coalition structures.” International Journal of Game Theory, 3, 217–237.
Boehm, C. (1999). Hierarchy in the Forest: The Evolution of Egalitarian Behavior. Harvard University Press.
Cacioppo, S., Frum, C., Asp, E., Weiss, R.M., Lewis, J.W., & Cacioppo, J.T. (2013). “A quantitative meta-analysis of functional imaging studies of social rejection.” Scientific Reports, 3(1), 1-3.
Dunbar, R. I. (1998). “The social brain hypothesis.” Evolutionary Anthropology, 6(5), 178-190.
Dunbar, R. I. (2004). “Gossip in evolutionary perspective.” Review of General Psychology, 8, 100–110.
Eisenberger, N. I., Lieberman, M. D., & Williams, K. D. (2003). “Does rejection hurt? An fMRI study of social exclusion.“ Science, 302(5643), 290-292.
Haidt, J. (2012). The Righteous Mind: Why Good People Are Divided by Politics and Religion. New York: Pantheon.
Harcourt, A. H., & de Waal, F. (Eds.). (1992). Coalitions and Alliances in Humans and Other Animals. Oxford: Oxford University Press.
Kalai, E., & Samet, D. (1987). “Weighted Shapley values.“ Games and Economic Behavior, 1(1), 54-63.
Soukeník, K. (2001). “Shapley value predicts the distribution of EU expenditure among member states.“ Journal of European Integration, 23(2), 165-198.
Shapley, L. S. (1953). “A value for n-person games.“ In H. W. Kuhn & A. W. Tucker (Eds.), Contributions to the Theory of Games, Volume II (pp. 307–317). Princeton: Princeton University Press.
Tooby, J., & Cosmides, L. (1996). “Friendship and the banker’s paradox: Other pathways to the evolution of adaptations for altruism.“ Proceedings of the British Academy, 88, 119-144.
Tooby, J., & Cosmides, L. (2010). “Groups in mind: The coalitional roots of war and morality.“ In Human Morality and Sociality: Evolutionary and Comparative Perspectives (pp. 191-234).
Von Hippel, W. (2018). The Social Leap: The New Evolutionary Science of Who We Are, Where We Come From, and What Makes Us Happy. Harper.
Von Neumann, J., & Morgenstern, O. (1944/2007). Theory of Games and Economic Behavior: 60th Anniversary Commemorative Edition. Princeton: Princeton University Press.
Walker, R. S., & Bailey, D. H. (2013). “Body counts in lowland South American violence.“ Evolution and Human Behavior, 34(1), 29-34.
There is some uncertainty on whether Benjamin Franklin actually said it.
Ethnographic studies of traditional societies in South America suggest that violence accounted for about 30% of adult deaths, mostly through raids and ambushes (Walker and Bailey (2013).
The idea of friends as protective allies has long been recognised. For instance, the Bible states:
A faithful friend is a strong defence: and he that hath found such an one hath found a treasure. (Ecclesiasticus 6:14-17).
The original study on social exclusion became famous for linking it to brain activation patterns similar to those of physical pain. However, later meta-analyses suggest this relationship is more complex (Cacioppo et al., 2013).
Man is by nature a political animal (zoon politikon). And he who is without a city-state, not by luck but by nature, is either a beast or a god. (Aristotle, 350 BCE)
The result also requires a more technical principle: Additivity—if two games are combined, a player’s total payoff should equal the sum of their payoffs in each individual game.
Intuitive explanation: The Shapley value is calculated by considering all the possible ways a person can be added to a coalition across all possible player orderings. In this example, there are six possible orders to form the coalition of three players: ABC, ACB, BAC, BCA, CAB, and CBA. What is A’s Shapley value?
If A is first, his added value is 1. This occurs in the two orders starting with A (ABC and ACB). When B is first, adding A contributes 4 (one case: BAC). When C is first, adding A contributes 3.5 (one case: CAB). When A is last, his added value is 5 (two cases: BCA and CBA). Across all orders, the average contribution of A is (1+1+4+3.5+5+5)/6 = 3.25. The same calculations yield 3 for B and 3.75 for C. Thus, if they form the grand coalition ABC, using the Shapley value to allocate the total payoff of 10, the split would be A (3.25), B (3), and C (3.75)—a nearly, but not entirely, equal distribution.
For mathematically inclined readers, here is the formal definition of the Shapley value as found in game theory textbooks. In a group of N players, if v(S) is the payoff for a given coalition S, the Shapley value of a player i can be written as:
The null player principle may seem harsh—if someone contributes nothing, they receive nothing. But as Tooby and Cosmides (1996) argue, evolutionary pressures likely selected against coalition partners who could not contribute:
To the extent an individual is in social relationships in which the assistance she delivers to her partners could easily be supplied in her absence by others, then there would be no necessary selection for her partners to help her out of difficulty. A 'replaceable' person would have been extremely vulnerable to desertion. - Tooby and Cosmides (1996)
Note that the Shapley value considers only internal exit options (e.g. forming a new startup with colleagues), not external ones (e.g. leaving for another firm composed of other players).
The Shapley value should not be taken as the one and only true way of splitting the gains from a coalition. Instead, it is simply a useful benchmark. It assumes equal probabilities of coalition formation and a fixed coalition payoff. In reality, different individuals have varying likelihoods of joining coalitions, and coalition success depends on members’ investments. Some extensions of the Shapley value relax these assumptions (Aumann and Drèze 1974, Kalai and Samet, 1987).
An empirical study has for instance found suggestive evidence that the Shapley value represents well the allocations of EU funds between European countries (Soukenik, 2001).
I have often wondered if the word "social" really applies to human beings, given, as you have noted, that we are quite a bit less social than truly social animals like ants.
Maybe we should use the term "coalitional" for humans instead of social? (Or is there another term for it?) At least I find it a lot more appropriate and even sets my expectations more correctly.
Thanks for a very informative article.
Thanks for this article, this subject seems never-ending in its applications. After all, as you say, our civilizations are essentially different networks of coalitions.
This makes sense because a human on his own is basically a very easy prey. We are not particularly fast or strong, we can’t fly or camouflage and we don’t have poison defenses or even a shell. Our only strength is that we can coordinate and communicate our shared knowledge and transmit it. But that turns out to be enough to basically win the Game of Life!