Coalitions, conflict and institutions

Picture — in round numbers drawn from Zimbabwe's 2008 election — a parliament split almost exactly in two. One party holds roughly a hundred seats, its great rival ninety-nine, and a third party, almost an afterthought, holds ten. Neither giant can govern alone. Whoever the ten-seat party sits with, wins — the arithmetic that forced the 2009 Government of National Unity. The ten-seat party is not a rounding error; it is, in a precise sense you will be able to compute by the end of this module, one third of the government. This module gives you the machinery to say exactly how much power it holds — and to see why the honest answer is not ‘ten seats out of two hundred and nine’.

Earlier modules modelled players choosing strategies in isolation and asked what no one would unilaterally abandon: the Nash equilibrium. Cooperative, or coalitional, game theory changes the primitive. It does not ask what strategy you play; it asks which coalition you join and what that coalition is worth. Binding agreements are assumed possible, so the interesting threat is no longer deviation by an individual but defection by a group — who can walk out, form a bloc, and do better on their own.

The characteristic function

A coalitional game with transferable utility is a pair (N, v). N is the set of players. The characteristic function v assigns to every coalition S ⊆ N a number v(S): the worth that S can guarantee itself, whatever the players outside S do. By convention v(∅) = 0, and the grand coalition N — everyone together — is worth v(N), the whole pie to be divided. Most games of interest are superadditive: two disjoint coalitions are worth at least as much merged as apart, v(S ∪ T) ≥ v(S) + v(T). There is then always a reason to build the grand coalition; the entire fight is over how to split what it is worth.

A voting game is the sharpest special case. Here v(S) takes only two values: 1 if S can pass what it wants — a winning coalition — and 0 if it cannot. A weighted voting game writes this with a quota q and weights wᵢ: coalition S wins when its combined weight reaches the quota, ∑ over i in S of wᵢ ≥ q. Seats in a chamber with a majority threshold are exactly such a game. Take our stylised parliament: three parties T, Z and M with weights 100, 99 and 10, a chamber of 209, and a simple-majority quota of q = 105.

Coalition S     Weight w(S)     Winning? (q = 105)     v(S)
------------------------------------------------------------------
{ }                  0           no                      0
{T}                100           no                      0
{Z}                 99           no                      0
{M}                 10           no                      0
{T, Z}             199           yes                     1
{T, M}             110           yes                     1
{Z, M}             109           yes                     1
{T, Z, M}          209           yes                     1

Read the table and one fact jumps out: every pair of parties is a winning coalition, and no party wins alone. In a precise sense the game is symmetric — what matters is not how many seats you bring but that any two of the three cross the line together. Hold that thought; it is the whole story of power indices.

The core: which splits survive?

Suppose the grand coalition forms and must divide v(N). An allocation x = (x_T, x_Z, x_M) is a proposed division with x_T + x_Z + x_M = v(N). When will it hold? Cooperative theory's first stability notion is the core: the set of efficient allocations that no coalition can block. Coalition S can block x when it can secure strictly more for its own members by walking out — when ∑ over i in S of xᵢ is less than v(S). An allocation lies in the core when no such coalition exists: for every S, the members already receive at least what they could grab by leaving.

Apply this to the parliament, with the prize — cabinet posts, ministries, the budget — normalised so that v(N) = 1 and every winning coalition is worth 1. The pair {T, Z} can secure the whole prize, so the core demands x_T + x_Z ≥ 1, which forces x_M ≤ 0. By the identical argument, {T, M} forces x_Z ≤ 0 and {Z, M} forces x_T ≤ 0. Three shares that must each be at most zero cannot possibly sum to one. The core is empty. There is no division of the spoils that some winning pair cannot overturn by dumping the third partner and splitting the whole prize between the two of them.

The Shapley value: power as marginal contribution

If the core can be empty, we still want a principled single answer to the question: what is each player worth? Lloyd Shapley's 1953 solution fixes four reasonable axioms — efficiency (the whole pie is shared out), symmetry (players who contribute identically get the same), the null-player property (a player who adds nothing to every coalition receives nothing), and additivity across games — and proves that exactly one allocation satisfies them. It has a beautifully concrete description. Imagine the players arriving one at a time in a random order. Each arrival is handed his marginal contribution — the amount by which he raises the worth of the coalition already in the room, v(S ∪ {i}) − v(S). The Shapley value φ(i) is that marginal contribution averaged over all n! possible orders of arrival.

Computing it: a stylised Government of National Unity

In a voting game the marginal contribution v(S ∪ {i}) − v(S) is either 0 or 1: it equals 1 exactly when i turns a losing coalition into a winning one — when i is pivotal. Averaging over orderings, the Shapley value collapses to something you can count: the fraction of all orderings in which the player is pivotal. That special case is the Shapley–Shubik power index. With three parties there are 3! = 6 orderings; we simply list them, and in each mark the party whose arrival first pushes the cumulative seat count up to the quota of 105.

Arrival order     Seats accumulating as each party enters       Pivot
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T, Z, M           100  →  199*  →  209                            Z
T, M, Z           100  →  110*  →  209                            M
Z, T, M            99  →  199*  →  209                            T
Z, M, T            99  →  109*  →  209                            M
M, T, Z            10  →  110*  →  209                            T
M, Z, T            10  →  109*  →  209                            Z
--------------------------------------------------------------------------
Pivot tally:      T = 2/6,     Z = 2/6,     M = 2/6
Shapley–Shubik power   φ = ( 1/3 ,  1/3 ,  1/3 )
Seat share             s ≈ ( 0.48 ,  0.47 ,  0.05 )
* = the entry that first reaches the quota.

Each party is pivotal in exactly two of the six orderings, so the Shapley–Shubik index is (1/3, 1/3, 1/3). Now set that beside the seats. T holds 100, Z holds 99, M holds 10 — and all three wield identical power. The ten-seat party is not one-tenth as powerful as the hundred-seat party; it is exactly as powerful. Power is not seat share. Power is the probability of being the party that tips a coalition over the line, and on that measure a kingmaker and a giant can be worth precisely the same. This is the formal content of the word ‘kingmaker’: it is why small parties in hung parliaments extract cabinet posts and policy concessions out of all proportion to their size.

Two indices: Shapley–Shubik and Banzhaf

Two power indices dominate the literature and it is worth knowing how they differ. The Shapley–Shubik index, which we just computed, counts orderings: of all the ways the players could line up, in what fraction is i the pivot? It inherits the Shapley axioms and treats every arrival order as equally likely. The Banzhaf index counts something subtly different — swings. It ignores order and asks: across all coalitions, in how many is i a swing voter whose departure would flip the coalition from winning to losing, expressed as a fraction of all such swings? The two indices usually rank the players the same way and occasionally disagree on the exact numbers; Shapley–Shubik is standard in economics, Banzhaf in law and constitutional design. Both carry the same lesson: to measure power, count pivots, not seats.

When bargaining fails: conflict

Cooperative theory assumes that binding agreements can be reached. Politics often cannot make that assumption, and the extreme case is war. Start from the puzzle James Fearon made canonical in 1995. Fighting is costly and its outcome uncertain; whatever a war eventually decides, there was almost always a peaceful settlement both sides would have preferred to the gamble of fighting — the same expected division, minus the destruction. War is ex-post inefficient. So the real question is not why interests conflict — they always do — but why rational actors can fail to reach the settlement that dominates fighting. Three answers survive scrutiny.

1. Private information with incentives to misrepresent. Each side knows its own strength and resolve; the other can only guess. Both have reason to overstate — to bluff for a better deal — so neither can credibly reveal the truth by talking. Fighting becomes the costly signal that words cannot send.
2. Commitment problems. A settlement is a promise about future behaviour, and there may be no way to make the promise binding. If today's concession shifts the balance of power, the side that gains has every reason to renege tomorrow — and both sides foresee it. When power is expected to move (a rising faction, a resource windfall, a disarmament) the declining side may prefer to fight now rather than accept a deal the other cannot be held to.
3. Issue indivisibilities. Some stakes resist splitting: a sacred site, a capital city, the presidency itself. If the prize cannot be divided or compensated in cash, the bargaining range that would make peace mutually acceptable may simply fail to exist.

The commitment problem is the sharpest lens on civil-war onset. A rebel movement and a government negotiate peace; the deal asks the rebels to disarm and rejoin ordinary politics. But disarming shifts power decisively to the state, and nothing binds the state — once its rivals have laid down their weapons — to honour the amnesties, the power-sharing, and the army integration it promised. Both sides can see this in advance. So the rebels rationally refuse to disarm, the government cannot credibly guarantee the future, and a settlement both would prefer to renewed war goes unsigned. This is why so many African peace agreements collapse at the implementation stage rather than at the negotiating table: the terms are agreed; what is missing is anything that can make today's concession bind tomorrow. Regional and international guarantors — an AU or UN force, an IGAD or ECOWAS mediation — are, in this light, attempts to manufacture exactly the commitment the parties cannot supply for themselves.

Institutions as self-enforcing equilibria

That last move — needing something to make a promise bind — is the doorway to the theory of institutions. An institution is not merely a rule written on paper; a rule that no one has an incentive to follow is worth nothing. An institution that works is a self-enforcing equilibrium: a pattern of behaviour such that, given that everyone else complies, no single actor gains by deviating. It persists for the same reason a Nash equilibrium persists — unilateral deviation does not pay — except that now the ‘strategies’ are things like ‘respect the election result’, ‘hand over power at the end of the term’, or ‘do not expropriate the merchants’.

Two literatures make this precise. Douglass North and Barry Weingast, studying England after 1688, asked how a sovereign can credibly promise not to expropriate — not to renege on debts, seize assets, or tax at will. A bare promise is not credible; the crown can always break it. What changed in 1688 was institutional: parliamentary control of the purse and independent courts raised the cost of reneging so high that keeping the promise became the crown's own best strategy. Credible commitment came not from good intentions but from a structure that made bad behaviour unprofitable — and the reward was a state that could suddenly borrow at rates its absolutist rivals could not touch. Daron Acemoglu and James Robinson turn the same key on democratisation: why would an elite ever concede power to those it could instead repress? Because a promise of future redistribution is worthless while the elite keeps the power to renege, extending the franchise institutionalises the concession — moving power itself rather than merely promising to use it kindly. Elites concede when the threat from below is credible, cheap repression is not available, and only a change in the rules, not a revocable favour, can buy peace.

The same logic explains one of the continent's most remarked-upon facts: the persistence of colonial borders. The lines drawn in European capitals in the 1880s cut through nations and fused rivals, yet at independence Africa's states, through the Organisation of African Unity in 1964, agreed to respect the frontiers they had inherited. Why keep arbitrary borders? Because mutual recognition is a self-enforcing equilibrium. Each government prefers to leave its neighbours' borders alone precisely because contesting them would license every neighbour — and every aggrieved group at home — to contest its own. The alternative to the arbitrary line is not a better line but a continent of competing claims with no natural stopping point. So the borders hold, not because they are just but because unilateral deviation from mutual recognition is ruinous for the deviator. As Jeffrey Herbst argued, Africa's states are sustained as much by this external equilibrium of recognition as by their internal capacity to broadcast authority over their own territory. An institution can be arbitrary in origin and iron in persistence, and the reason is always the same: no one gains by being the first to break it.