Coalition games
Formal models for social justice, colonization, elite capture, interest groups, and the like
2022-07-08 — 2026-04-30
Wherein divide-and-rule is reframed as a coalition game via the partition function form, and the British reorganisation of India after 1857 is taken as the worked example.
Assumed audience:
Insurgents and conquerors alike
Content warning:
War, colonization, elite capture, and the like.
Placeholder for remarking on particular types of coalition games.
I’ve heard enough people fulminating about the other side recently that I want to dissect the fulminating dynamics for my own satisfaction the right way, which is to say: using maths.
I would ideally present these strategies without judgement or moralising. Here simply to regard the dynamics we can expect in coalition formation and to introduce terminology so we can understand the pathologies that arise.
These dynamics — building winning blocs and breaking opposition ones — turn up across two literatures that I think are kinda the same but don’t cite each other much.
The mathematics here is fun but before we get too practical with it, flagging some unrealistic parts of the models. The formalisms below are mostly full-information games. Players know who the others are, what each coalition is worth, what the payoffs of each outcome are, and they all maximise utility rationally. Out in the world, coalitions form via persuasion, signalling, deception, manufactured identities, and shifting public attention. The model below captures one structural mechanism — who-would-gain-from-which-arrangement — but in contemporary democratic politics much of the action is in the persuasion-and-signalling layers. The British did not divide India by explicitly solving a formal partition function game; they did it via centuries of patronage and cultivated divisions and it sounds like they started out doing it more-or-less by happenstance (Dalrymple 2022). Maybe they did eventually think about the colonial project this way? Who knows? Maybe from the inside coalition games feel like loyalty and betrayal and strife, but it still all comes down to a sociopath’s optimal manipulation of the partition function game. It is interesting nonetheless to consider how much this structural mechanism can explain, and to have a vocabulary for it.
1 Forming a bloc
The foundational result is Riker’s size principle (Riker 1962): in a zero-sum political game where what matters is winning, rational coalition-builders form minimum winning coalitions — just big enough to clinch the win, no bigger — because every extra member dilutes the spoils.
A worked example— Three voters, each with one vote, deciding how to split a prize of \(1\) by majority rule. The winning coalitions are \(\{1,2\}\), \(\{1,3\}\), \(\{2,3\}\), and \(\{1,2,3\}\). Riker’s prediction is that a rational proposer forms a two-person coalition (any of the three pairs) and shares the prize between just those two; the third voter, not needed for the majority, gets nothing. Forming the grand coalition would split the prize three ways, leaving the proposer worse off.
The non-cooperative version of the same insight is Baron–Ferejohn legislative bargaining (Baron and Ferejohn 1989): a randomly-chosen proposer offers shares to just-enough partners to clear a majority, captures most of the surplus, and any partner who declines gets replaced by another.
The cooperative complement is the coalition-structure value. Once the partition into blocs is given, the Shapley value generalises to a value that respects the partition (Aumann and Dreze 1974; Owen 1977). This answers the question “given that these coalitions formed, who gets what?” (as opp. “which coalitions form?”, as in the earlier models)
2 Breaking a bloc
The dual question — keeping opposition coalitions small enough to be unthreatening — has a younger, mostly political-economy literature. Famous formal models include Acemoglu, Robinson, and Verdier (2004) and Padró i Miquel (2007): a ruler stays in power by selectively transferring resources between subgroups so that no two of them have aligned interests against her. Even-handed treatment of subjects is destabilising; uneven treatment is stable. The British-in-India dynamics covered below are perhaps an instance — selectively favouring some princely states, some castes, some communities over others, with the favours timed to break up coordination potential.
Contest theory (Hirshleifer 1991b, 1995) supplies the underlying economics. Agents have a fixed budget of effort, which they can spend either on production (making wealth) or on conflict (fighting over wealth that already exists). Conflict consumes effort without creating output: when two rivals fight, both come out poorer. A dominant party who can stay out of the fighting, and who can encourage her rivals to fight one another, reaps the benefit of their depletion at no cost to herself.
The same problem appears in mechanism design, labelled differently. Imagine shareholders who employ a manager to monitor a worker. The manager observes the worker’s effort and reports to the shareholders, who pay both. The shareholders worry that the manager and worker collude — the manager files an inflated effort report in exchange for the worker kicking back a share of the inflated wage. The optimal contract here has to be collusion-proof (Tirole 1986): it has to leave no pair of agents jointly better off by colluding than by reporting truthfully. This is generally costlier than the case where collusion is impossible, because the principal has to pay each agent more — a collusion premium — to keep them honest. If the shareholders can additionally play the manager and worker against each other, paying each more when their reports disagree, they can recover some of that premium. The structure is identical to political divide-and-rule, just bounded by the firm rather than the polity: a principal who can keep her subordinates’ interests misaligned pays less for their cooperation than one who cannot.
3 Coalition formation
Both forming and breaking are special cases of a more general question: given a coalitional game, which partitions \(\pi\) are stable? Coalition formation theory supplies the formalism. Two specifications come up most often: the partition function form, where coalitions have numerical worths that depend on the surrounding partition, and hedonic games, where each player has an ordinal ranking over the coalitions she might end up in.
3.1 Partition function form
A standard cooperative game is specified by a characteristic function \(v(S)\): the worth of coalition \(S\) in isolation. This formulation is silent about how \(S\)’s prospects depend on what the other players are doing. Whether the rest are unified or fragmented, \(v(S)\) stays the same — so a characteristic-function game has nothing to say about divide-and-rule.
The partition function form (Weber 1994) generalises \(v(S)\) to \(v(\pi, S)\), where \(\pi\) is the partition of \(N \setminus S\) into other coalitions. A coalition’s worth now depends on the structure of its environment, and divide-and-rule has a place to live: the ruler arranges the world so her opposition’s partition is the one she likes best.
Time for a worked example! Take four players: three opposition figures plus a ruler, \(N = \{1,2,3,r\}\). The ruler’s solo coalition \(\{r\}\) has worth \(v(\pi, \{r\})\), where \(\pi\) partitions \(\{1,2,3\}\). Fragmented into singletons, the ruler picks them off in turn and \(v\) is high. United as \(\{1,2,3\}\), the ruler is overwhelmed and \(v\) is low. Same players, same individual capabilities, different prospects for the ruler depending on how the others are partitioned. Divide-and-rule is the ruler manipulating payoffs (selective bribes, threats, propaganda) so that the singleton partition becomes stable instead of the unified one.
3.2 Hedonic games
A different specification: drop the value function and give each player a ranking over coalitions she could end up in (Drèze and Greenberg 1980; Bogomolnaia and Jackson 2002). Useful when payoffs are qualitative — flatmates, research groups, religious communities — and there is no natural numerical measure of a coalition’s “worth,” only of how each member ranks the alternatives.
Even simple hedonic games can fail to have a stable matching. The classical roommate problem (Cechlárová and Romero-Medina 2001) supplies preference profiles where every pairing of four students into two flats leaves someone wanting to swap.
3.3 Stability and the action question
For either formalism, stability is the question of whether a partition \(\pi\) is one no subset of players would profitably deviate from (Yi 1997). Stability concepts differ in what counts as a profitable deviation, and in what deviators believe the rest will do in response. This is the positive question — given a game, which \(\pi\) holds together.
The action question — how should a ruler manipulate the game so her preferred \(\pi\) is stable? — is less developed. Acemoglu, Robinson, and Verdier (2004) and Padró i Miquel (2007) are specific political-economy instances; Bartholdi, Tovey, and Trick (1989) and the Banzhaf and Shapley–Shubik power indices (Shapley and Shubik 1954) supply pivotality measures useful for “who to bribe first.” A general theory of optimal manipulation of partition function games, AFAICT, does not exist.
A loose end: once a partition has formed, allocating spoils within each coalition uses the Shapley value, generalised to take \(\pi\) as a parameter (Aumann and Dreze 1974; Owen 1977; Bruner and O’Connor 2017) — the coalition-structure value from Forming a bloc.
Bloc-forming theory is mostly written from the inside (“how do we coordinate?”), bloc-breaking from the outside (“how does the principal prevent coordination?”). They look to share the same formalism.
For broader surveys: Ray (2007), Ray and Vohra (2015), Greenberg (1994), and Weber (1994).
4 Divide and rule
“Divide and rule” is the casual, applied label for the strategy that the previous section called the action question — choosing \(v(\pi, S)\) to make a preferred \(\pi\) stable. The Latin maxim divide et impera is older than the British Empire; Caesar, Machiavelli, and Louis XI all variously get the attribution. The British in India are the textbook case partly because they were so obliging as to use the phrase themselves.
The framework cuts both ways. As readers of the colonial archive we gain a vocabulary: army-by-caste, princes-by-treaty, electorates-by-community become legible as moves in a partition-function game. And the administrators themselves seem to have reasoned in something close to those terms, informally, well before the formalism existed. How tight that mapping is — whether their thinking would survive any honest formalisation — is a different and probably less interesting question than whether the lens illuminates anything.
The most familiar attestations come from after the 1857 Mutiny:
Divide et impera was the old Roman maxim, and it should be ours.
— Lord Elphinstone, Governor of Bombay, to the post-Mutiny Commission, as quoted in Streets (2004). The Punjab Committee’s 1858 report on army reorganisation invoked the same principle by name. A later memorandum from Brigadier John Coke is more explicit still: the existing separations between religions and races should be upheld in full force, not amalgamated.
The shape of the reasoning is recognisable in administrators’ own words, even though none of them was working \(v(\pi, S)\) out on paper.
4.1 The administrative record
After 1857 the Bengal Army was reconstituted along caste and regional lines so that no single grouping could coordinate a second mutiny (Streets 2004). The martial races doctrine — recruit Sikhs, Punjabis, Gurkhas; exclude the castes and regions that rose in 1857 — kept the army’s most capable units economically dependent on the Raj. The princely-state system left roughly two-fifths of the subcontinent’s land area in the hands of nominally sovereign rulers, each treaty-bound to the British and balanced against the others. Separate electorates from the Morley–Minto reforms (1909) and the Communal Award (1932) gave Hindus and Muslims separate ballots and reserved seats, formalising the categories the British wanted to manage. The census, beginning in earnest in 1872 and especially from 1901, fixed fluid local identities — caste, sub-caste, religion — into administrable categories. Dirks (2001) argues that the colonial state did not invent these distinctions but systematised them; (cf seeing like a state. Bayly (1996) traces how British intelligence-gathering produced the categorical knowledge that later policy operated on.
These arrangements are what the partition function form makes legible: the Raj’s prospects depend on \(\pi\) — the partition of the army, the princes, the electorates, and the census categories — in the way \(v(\pi, \{r\})\) depends on \(\pi\).
4.2 How calculated was it?
Dalrymple (2022) makes the “cock-up case” for the East India Company era: 1757–1803 was opportunism, not strategy — no centralised divide-and-rule policy because there was no centralised policy. Bayly (1996)’s view of the late EIC and early Raj is intermediate: a sophisticated information-state that read local rivalries and exploited them, but reactively. The 1857 Mutiny is an inflection point — afterwards the explicit quotes appear, and the institutional machinery becomes more deliberately fragmentation-oriented (Streets 2004). Jalal (1994) complicates this: the British thought separate electorates would manage communal balance, but the institutional logic ran the other way, eventually producing the Hindu-Muslim polarisation that ended in the literal Partition.
Whether the divide-and-rule arrangements were calculated, improvised, or drifted into, the partition function form describes their structure once they are in place. The lens does its work either way.
4.3 Popular treatments
5 Homework exercise
I care about this not just as a historical exercise. Left-right polarization, ethnic scapegoating, the manufactured culture-war binary — all bear a suggestive resemblance to the shape of the ruler’s \(v(\pi, \{r\})\), with r played not by a single ruler but by whoever benefits from everyone else being partitioned along a useful axis: a party, a media organisation, a platform, a coalition of donors. Anyone whose payoff goes up when the rest of the population fragments has a partition-function game to play.
Left-right polarization in two-party democracies is the obvious instance: the binary partition is structurally stable for both incumbents, since alternative partitions (multi-party coalitions, cross-cutting cleavages) would unbind them.
Ethnic scapegoating perhaps fits the Contest theory mould: the ruler does best when the in-group is unified around a target, so a stable partition is everyone-vs.-them. Anti-immigrant politics across Western democracies, the Hindu-Muslim axis in contemporary India, historical European antisemitism — different cases, same template.
Othering and identity-formation extend Dirks’s census argument: a category named and counted often enough acquires its own politics. The “Hispanic” census category, added in 1970, has hardened over decades into a coalition with its own electoral arithmetic. Once a partition is administered, the partition-function game starts to play itself.
I would not push the analogy too hard. Contemporary politics has many moving parts the formalism doesn’t speak to — as mentioned above, persuasion, signalling, manufactured identities, media incentives, the unrealistic-formalism warning above. But it is a lens worth pointing at the present. The fulminating that opened this post is itself a move in someone’s partition-function game; it pays to ask whose.
6 Incoming
I feel this relates to Critch’s infamous boundaries programme, e.g. «Boundaries», Part 1: a key missing concept from utility theory, but am reluctant to do the work of making that correspondence precise.