Most of the strategic situations you will actually meet in African markets and policy are not settled in a single simultaneous move. An entrant decides whether to challenge Safaricom, and only then does Safaricom respond. A finance minister announces a fiscal rule, and only then do lenders reprice the debt. A central bank picks an exchange-rate regime, and only then do households and traders form the expectations that make or break it. The order of play matters, and so does what each side can see when it moves. Once you take timing seriously, something striking happens to threats: many of the fiercest ones — the price war that will crush any entrant, the austerity that will surely follow a bailout — turn out to be bluffs that no rational player would ever carry out. This module gives you the apparatus to tell a real threat from an empty one, and to see why deliberately tying your own hands can be the most powerful move on the board.
The extensive form: games as trees
The extensive form represents a game as a tree. Each internal node is a decision point owned by exactly one player; the branches leaving a node are the actions available there; and each terminal node — a leaf — carries a payoff vector, one number per player, listing what everyone receives if play reaches that outcome. Play starts at the root (the first mover) and travels down a single path to a leaf. Where the normal form — the payoff matrix you met earlier — records only who chooses what, the extensive form also records when they choose and what they know when they do. Those are the two ingredients that simultaneous-move analysis throws away.
Reading a game tree
A tree has four parts. Nodes are decision points, each labelled with the player who moves there. Branches are actions. Leaves are outcomes, each tagged with a payoff vector written in a fixed player order, e.g. (Entrant, Incumbent). The root is where play begins. One refinement you will need: an information set is a group of nodes a player cannot tell apart when moving. If every information set is a single node — everyone always knows exactly where they are in the tree — the game has PERFECT information, and backward induction applies cleanly. If some information set contains several nodes (a player must move without observing a rival's earlier choice), information is IMPERFECT, drawn as a dashed line joining those nodes. Every game in this module has perfect information unless stated otherwise.
Backward induction
Backward induction is the engine that solves a perfect-information tree. The logic is “look forward, reason back.” You start at the decisions closest to the leaves — the last movers — and ask what a rational player does there. You keep that best action, replace the subtree hanging below it with the payoff it produces, and treat that payoff as if it were a leaf. Then you step up to the second-to-last movers and repeat. Working from the tips of the tree back to the root, you fold the whole game down to a single predicted path. In any finite game of perfect information this process always terminates and, generically, delivers a unique prediction — a result that traces back to Zermelo and to Kuhn.
- Identify the last decision nodes — those whose branches all lead directly to leaves.
- At each, select the action that maximises the moving player's own payoff. Record it; discard the rest.
- Replace that node with the payoff vector of the action you kept. The node is now effectively a leaf.
- Move up to the next layer of decisions and repeat, using the folded-in payoffs.
- Continue until you reach the root. The retained actions, read top to bottom, are the equilibrium strategies; the path they trace out is the predicted outcome.
Subgame perfection: ruling out empty threats
Backward induction is more than a solving trick; it defines a solution concept. A subgame is any node together with everything that hangs below it — a self-contained smaller game. A strategy profile is a subgame-perfect equilibrium (SPE) if it prescribes a Nash equilibrium in every subgame, not merely in the game as a whole. In a finite game of perfect information, the strategies backward induction produces are exactly the SPE. This matters because SPE is a refinement of Nash: it discards Nash equilibria that are propped up by plans a player would never actually execute if the relevant node were reached. Those plans are non-credible threats, and the whole point of subgame perfection is to delete them.
Subgame perfection and the credibility test
SPE: a strategy profile that induces a Nash equilibrium in every subgame — including subgames that equilibrium play never actually reaches. The credibility test that falls out of it: a threat, or a promise, is credible only if carrying it out is a best response at the node where it would actually have to be executed. If, on arriving at that node, the player would rather do something else, the threat is empty — and no rational opponent should be deterred by it. Nash equilibrium can be sustained by empty threats; subgame perfection cannot.
Worked example: Safaricom and the entrant
Take the most familiar contest in East African markets: a would-be challenger weighing entry against Safaricom, the dominant incumbent whose subscriber base and M-PESA rails define the terrain. Model it as a two-stage game of perfect information. First the Entrant chooses Stay Out or Enter. If it enters, Safaricom observes the entry and chooses Fight — a price war: aggressively cut tariffs, zero-rate transfers, bundle data — or Accommodate, settling into a shared market. The payoffs below are stylised and ordinal: they encode preference rankings, not shillings, and nothing in the argument depends on their exact magnitudes.
Rank the outcomes from each side. The Entrant most prefers to enter and be accommodated (a viable slice of a large market), is content to stay out (zero — its outside option), and least wants to enter into a price war that burns its thin capital. Safaricom most prefers that the entrant stays out and leaves the monopoly intact; failing that, it prefers a quiet shared duopoly to a price war — because a war forces it to slash prices across its own huge existing base, and it has far more revenue to defend than the newcomer does. That last point is the crux: fighting is costly for the incumbent precisely because the incumbent is large.
Entrant moves first:├─ Stay Out ────────────────→ (0, 10) Safaricom keeps its monopoly└─ Enter → Safaricom moves:├─ Fight (price war) → (-3, 3) both bleed; the entrant is driven out└─ Accommodate → (4, 6) the market is shared
Solve it by backward induction. Begin at Safaricom's node, the only decision after Enter. Fighting yields 3; accommodating yields 6; since 6 > 3, Safaricom accommodates. Fold that in: the Enter branch is now worth (4, 6). Step up to the Entrant's decision. Staying out yields 0; entering now yields 4, because the Entrant correctly foresees accommodation. Since 4 > 0, it enters. The subgame-perfect equilibrium is therefore Enter, then Accommodate, with outcome (4, 6). Entry happens.
The threat that isn't
Safaricom might declare, loudly and often, “we will fight any entrant to the death.” Suppose the Entrant believed it. Then entering would mean a payoff of -3 against a stay-out value of 0, so the Entrant would stay out and Safaricom would keep its monopoly worth 10. Notice that this pair — Entrant stays out, Safaricom plans to fight — is a Nash equilibrium of the whole game: given the threat, staying out is a best response, and given that the entrant stays out, Safaricom's plan is never put to the test. But it is NOT subgame perfect. At the only node where fighting would actually be executed, fighting pays 3 and accommodating pays 6. Safaricom would not carry out the threat. Backward induction deletes this equilibrium automatically — which is exactly what a refinement is for.
Making the threat credible: commitment
The incumbent's problem is not a shortage of menace; it is that, once entry has happened, it would genuinely rather accommodate. To deter entry it must change its own future payoffs so that fighting becomes its best response — and, crucially, do so visibly and irreversibly before the entrant decides. Three commitment technologies do this. First, sink capacity or cost: build ahead of demand, or invest in a very low-cost network, so that a price war is cheap to wage and idle capacity makes accommodation wasteful — the sunk outlay re-orders the payoffs at the Fight node. Second, build a reputation: an incumbent facing a sequence of potential entrants that fights the first one, at a loss, can deter all the rest, so fighting pays in the repeated game (the chain-store logic). Third, write binding contracts: a public price-match guarantee, or most-favoured-customer clauses, that legally force a price cut and make accommodation costly.
Entrant moves first:├─ Stay Out ────────────────→ (0, 8) monopoly, net of the sunk capacity cost└─ Enter → Safaricom moves:├─ Fight (price war) → (-3, 5) pre-built capacity makes fighting cheap└─ Accommodate → (4, 3) idle capacity is now wasted
Re-solve Figure 2. At Safaricom's node, fighting now yields 5 against accommodation's 3, so it fights. The Entrant, foreseeing -3 versus a stay-out value of 0, stays out. The outcome is (0, 8) — and here is the payoff to strategy. By sinking money into capacity it may never use, Safaricom raised its own equilibrium payoff from 6 (accommodating an entrant) to 8 (deterring entry outright). It improved its position by removing its own freedom to back down. That is the paradox at the heart of commitment: fewer options can be worth more than more options — but only if the commitment is observable (the entrant must see the capacity) and irreversible (it cannot be quietly undone once the entrant has committed).
Commitment and first-mover advantage: the general principle
The same logic runs underneath the classic first-mover advantage. In Stackelberg's sequential quantity game, the leader sets output first, knowing the follower will best-respond along its reaction function; by moving to a point the follower must then accommodate, the leader captures more than it would in the simultaneous Cournot game. The advantage does not come from moving early as such — it comes from moving in a way that is observed and cannot be reversed, which is what lets the early move reshape the opponent's problem. Read carefully, first-mover advantage is commitment advantage. An early move the rival cannot see, or that you can costlessly revise, changes nothing.
Burning bridges
The oldest version of this idea is military. A commander who burns the bridge behind his own army has destroyed his option to retreat — and, by doing so, convinces the enemy he will fight to the end, which can win the battle without one. Cortés scuttling his ships and Sun Tzu's advice to fight on “death ground” are the same move: throwing away an option as a source of power. Thomas Schelling formalised it for economics — the power to constrain an adversary may rest on the power to bind oneself. But the caveat is the whole game: a bridge you can secretly rebuild, a peg you can abandon overnight, a rule with an escape clause you control — these are not commitments at all. Only a burned bridge deters.
Where the model bites — and where it bends
Backward induction is a disciplined baseline, not a law of nature, and a serious analyst states its assumptions before trusting its verdict. It requires that the tree and the payoffs are common knowledge, that rationality is common knowledge (you are rational, you know the other side is, they know you know, and so on without end), and that players can actually compute the fold-back. In long trees that common-knowledge tower gets shaky, and laboratory play often departs from the prediction — subjects cooperate far into the centipede game, and reject unfair splits in the ultimatum game rather than take a positive payoff. Payoffs in the field are rarely known as cleanly as a diagram suggests, and reputations, fairness norms and repeated interaction all reshape them.
- Common knowledge of the tree — in real bargaining, parties often disagree about what options and outcomes even exist.
- Common knowledge of rationality — plausible for two sophisticated firms; strained across a crowd of heterogeneous voters or traders.
- Correct payoffs — the analysis is only as good as the rankings you feed it; a single mis-ranked node can flip the solution.
- Observability and irreversibility of commitments — a commitment nobody can see, or that can be reversed, does no strategic work.
- A finite horizon — the fold-back needs somewhere to begin; infinite-horizon games are handled by related but distinct tools.
A commitment device is only as good as its exit
Every commitment device in this module lives or dies on one question: how hard is it to reverse, and can everyone see that? A currency peg backed by thin reserves is a bridge the market knows you can be forced to abandon, which is precisely why speculative attacks come — the commitment is soft, so it invites the test. A fiscal rule with an escape clause the finance ministry can invoke at will is discretion wearing a rule's clothes. The design question is never merely “announce a rule”; it is “what makes the announcement binding” — constitutional entrenchment, external enforcement, reserves large enough to make the promise self-fulfilling, or a reputation too valuable to spend.
The two exercises below run this machinery through the other two African cases promised at the outset — a central bank's regime choice, and a sovereign at a debt wall. In both, the interesting move is a player binding itself; and in both, the method is identical: solve the tree from the leaves back, and ask at every node whether the promised action is one the player would truly take.
Central Bank chooses its regime:├─ Commit (hard peg / legislated rule / independent CB)│ expectations anchor Low, inflation stays Low → (0, -0.2)└─ Discretion → Public forms expectations:├─ Expect Low → Central Bank sets inflation:│ ├─ Low → (0, 0)│ └─ High → (-1, +1) surprise boom now, inflation later└─ Expect High → Central Bank sets inflation:├─ Low → (-2, -2) recession from surprise disinflation└─ High → (-1, -1) inflation entrenched, no boom
Exercise
Figure 3 models a central bank's credibility problem — a sequential rendering of the Kydland-Prescott time-inconsistency result. The bank first chooses its regime: Commit to a binding low-inflation framework (a hard peg such as the CFA franc's link to the euro, a legislated rule, or a genuinely independent central bank) or retain Discretion. Under discretion the public forms inflation expectations (Low or High), and the bank then sets actual inflation (Low or High); a surprise — inflation above expectations — buys a temporary output boom, while inflation itself is costly and disinflation below expectations causes a recession. Payoffs are (Society, Central Bank). (1) Solve the discretion subgame by backward induction and explain why the public rationally expects High inflation. (2) Explain precisely why the bank's promise of low inflation under discretion is an empty threat in the sense of this module. (3) Evaluate the Commit branch and state the subgame-perfect regime choice. (4) Name a real African commitment device in each mould, and the exit that could still break it.
Government faces a debt wall and moves first:├─ Default now ─────────────────────→ (-4, -6) access lost, no reform└─ Seek IMF programme → IMF moves:├─ Refuse → government defaults anyway → (-5, -6) worst of both, after delay└─ Offer (financing + conditionality) → Government moves:├─ Comply with conditions → (-2, -3) austerity now; restructuring; access restored└─ Renege (take cash, quit) → (-6, -7) programme collapses; tranches forfeited
Exercise
Figure 4 models a government at a debt wall — think of the distress that put Zambia into default in 2020, and Ghana into default and then an IMF programme in 2022-2023. The government moves first: Default now, or Seek an IMF programme. If it seeks one, the IMF (standing in for official and private creditors) chooses to Offer financing plus conditionality, or to Refuse; if refused, the government defaults anyway after a costly delay. If an offer is made, the government chooses to Comply with the conditions or to Renege — take the disbursement and abandon the reforms. Payoffs are (Government, Creditors). (1) Solve the tree by backward induction and state the subgame-perfect equilibrium. (2) Now suppose reneging carried no penalty, so its government payoff were -1 rather than -6. Re-solve, and show how the cooperative outcome unravels. (3) Explain how IMF conditionality — tranched disbursement, prior actions, benchmarks — restores the outcome in (1), and connect it to the commitment logic of this module. (4) Give the second, domestic reason a reformist government may actively want the programme.