From one decision-maker to many
Walk down almost any street in Nairobi, Kampala or Dar es Salaam and you will pass a row of mobile-money kiosks — a green M-Pesa awning beside a red Airtel Money one, sometimes sharing a wall. Behind those agents sit two firms setting transaction fees, and each sets its fee with one eye on the rival. If Safaricom cuts the charge on a small transfer, Airtel feels it within the week; if Airtel launches a promotion, Safaricom's product team is in a meeting by Monday. Neither firm can choose its best fee by staring only at its own costs and customers. It must anticipate the other. That mutual anticipation — not the size of the firms, not the technology — is what makes this a game.
Now picture a smallholder maize farmer in the Rift Valley deciding how many acres to plant. She, too, faces uncertainty: the rains may fail, and the price at the local market or the National Cereals and Produce Board may be high or low at harvest. But she is one of millions of maize growers, and nothing she plants will move the national price by a single shilling. When she chooses, she is not trying to out-guess a rival who is simultaneously trying to out-guess her. She is choosing against nature and against a market price she takes as given. Economists call her a price-taker, and her problem a decision problem: a single agent optimising against an environment that does not strategise back.
Strategic interdependence
A situation is strategic when your best action depends on what someone else does, and their best action depends on what you do. Each player must therefore reason about the others' reasoning. When no such interdependence exists — when the environment does not think back — you face a decision problem, not a game.
Hold the two side by side. The farmer's payoff depends on her own choice and on impersonal chance. Each mobile-money firm's payoff depends on its own choice and on the deliberate choice of a rival who is reasoning about it in turn. That second feature — strategic interdependence — is the entire subject of this course. Decision theory handles the farmer. Game theory is what you need the moment a second reasoning agent enters and their fate and yours become tangled together by the choices you both make.
The ingredients of a game
Every game is built from four ingredients; name them well and half your analysis is done. The first is the set of players — the decision-makers whose choices matter. In our pricing game the players are Safaricom's M-Pesa and Airtel Money. Choosing the players already involves judgement: we have left out the Central Bank of Kenya, the agents and the customers, treating them for now as part of the fixed environment. The second ingredient is each player's actions — the concrete moves available. Here we simplify a rich pricing decision down to two actions per firm: set a High fee or a Low fee.
The third ingredient needs care, because two words that sound alike — action and strategy — are not the same. An action is a single move. A strategy is a complete plan of action: a rule telling you what to do in every situation you might find yourself in. In a one-shot game where each player moves once and simultaneously, a pure strategy is just an action, and the two coincide. But consider a matatu deciding whether to enter a route a SACCO already controls, where the SACCO can watch and then respond. There, a strategy for the SACCO is not a single move but a contingent plan — for instance, accommodate the newcomer if it enters, do nothing otherwise. Keep the distinction: we will need it the moment games acquire timing.
The last two ingredients are outcomes and payoffs. An outcome is what results once every player has chosen — a combination of actions, such as (M-Pesa Low, Airtel High). A payoff is a number attached to each outcome, for each player, representing how much that player prefers it. This is the subtle one. Payoffs are not defined as money; they are a numerical stand-in for preferences over outcomes, and a well-drawn payoff already absorbs everything the player cares about — profit, yes, but also risk, market share, reputation and the shadow of the regulator. When we write a 10 in a cell we mean this player ranks this outcome here, not this player receives ten shillings.
A game in normal (strategic) form
A game in normal form is three things: a set of players i = 1, 2, …, n; for each player a set of strategies Sᵢ; and for each player a payoff function uᵢ that assigns a number to every combination of the players' strategies (s₁, …, sₙ). The payoff uᵢ measures how much player i likes that combination. Nothing else — no story, no history — is needed to define the game.
Payoffs are preferences, not shillings
A payoff is a number that ranks outcomes by how much a player prefers them; it already folds in everything the player cares about — money, yes, but also risk, reputation, fairness, the wrath of a regulator. Do not read the numbers as literal cash, and do not assume that one player's high payoff implies another's low one. Confusing payoffs with money — or assuming every game is a fight over a fixed prize — is the single most common beginner's error.
Reading a payoff bimatrix
For a two-player game with a handful of actions each, the whole game fits in one grid called a bimatrix, and learning to read it fluently is a basic skill. One player — call them the row player — chooses a row; the other, the column player, chooses a column. Their two choices together pick out a single cell, and inside that cell sits an ordered pair of numbers. The universal convention is: the first number is the row player's payoff, the second is the column player's. Read a cell aloud as row gets this, column gets that. That is all a normal-form game is — a way of tabulating who gets what for every combination of choices.
The M-Pesa vs Airtel Money pricing game
Let us build the bimatrix for the fee game. M-Pesa is the row player, Airtel Money the column player; each chooses a High or a Low transaction fee. We need payoffs that capture the strategic logic, so reason it through. If both keep fees High, they enjoy comfortable margins and split the market roughly as it stands — a good result for each. If both cut to Low, they trigger a price war: customers win, margins are thin, and each does poorly. The interesting cells are the mismatches. If one firm alone cuts to Low while the other holds High, the cutter poaches a wave of price-sensitive customers and does best of all, while the firm left holding a High fee is undercut and does worst. The numbers below — illustrative units, not real profits — encode exactly that ranking.
Airtel MoneyHigh fee Low fee+-------------+-------------+M-Pesa High fee | 10 , 10 | 2 , 14 |+-------------+-------------+M-Pesa Low fee | 14 , 2 | 5 , 5 |+-------------+-------------+
Read the grid. The top-left cell is both firms holding High: (10, 10) — each does well. The bottom-right is the price war: (5, 5) — each does badly. The off-diagonal cells are the mismatches: bottom-left is M-Pesa Low against Airtel High, paying (14, 2) — a bonanza for M-Pesa, a rout for Airtel — and the top-right mirrors it at (2, 14). Notice the ranking each firm faces, from best to worst: undercut alone (14), mutual restraint (10), mutual war (5), be undercut (2). That ordering — temptation above cooperation above mutual punishment above the sucker's outcome — is the fingerprint of a particular and famous kind of game, which we solve next.
Solving the game: dominance and the dilemma
Fix Airtel's choice and ask what M-Pesa should do. If Airtel sets a High fee (the left column), M-Pesa earns 10 by matching High and 14 by cutting to Low — so Low is better by 4. If Airtel sets a Low fee (the right column), M-Pesa earns 2 by holding High and 5 by cutting to Low — so Low is better by 3. Low beats High in both columns: cutting fees is a strictly dominant strategy for M-Pesa. The game is symmetric, so the identical argument runs down the two rows for Airtel. Both play Low, and the outcome is the bottom-right cell, (5, 5). Yet both would earn 10 at (High, High). The strategy each is compelled to choose leaves both worse off than the restraint each would have preferred to see. That gap between individually rational choice and a jointly better outcome is the engine of half of game theory.
Dominant strategy and Nash equilibrium (a preview)
A strategy strictly dominates another if it yields a higher payoff no matter what the opponents do; a rational player never uses a strictly dominated strategy. A Nash equilibrium is a profile of strategies — one per player — in which no player can raise their own payoff by changing strategy alone, holding the others fixed. We build the equilibrium concept carefully in later modules; for now, notice that (Low, Low) has this no-regret property, while (High, High) does not — from (High, High) either firm would gain by defecting to Low.
The trap is not stupidity
Each firm plays Low because Low is genuinely its best reply to anything the rival might do — that is exactly what makes the outcome so sticky. Escapes from such traps do not come from exhortation; they come from changing the game: playing it repeatedly so that punishment becomes possible, writing an enforceable contract, or having a regulator alter the payoffs. Hold on to that distinction — the game versus a single play of it — throughout the course.
Zero-sum or not: conflict versus mixed motives
Add the two payoffs in each cell of the fees game: (High, High) sums to 20, the two off-diagonal cells to 16 each, and (Low, Low) to 10. Because the total is not the same in every cell, the game is not constant-sum — it is non-zero-sum. The size of the pie changes with the choices, which is precisely why there is something to cooperate about: moving from (Low, Low) to (High, High) conjures 10 units of value out of thin air. Contrast a genuinely zero-sum encounter — a SACCO inspector deciding whether to audit a conductor who is deciding whether to pocket a fare. Every shilling the conductor hides is a shilling the SACCO loses; the two payoffs in each cell sum to the same constant, and there is pure conflict with nothing to jointly gain. Most of the interesting games in economics — trade, public-goods provision, tax competition between African states — are non-zero-sum, and treating them as pure conflict throws away the very thing worth analysing.
The working assumptions
Two working assumptions underpin the standard analysis, and you should state them out loud rather than smuggle them in. The first is rationality: each player has consistent preferences over outcomes, represented by the payoffs, and chooses the action that maximises their expected payoff given their beliefs about what the others will do. Rationality here is thin — it is about consistency between ends and means, not about selfishness or cold-heartedness. A player who cares deeply about fairness is perfectly rational; that concern simply shows up inside the payoffs. The second assumption concerns not one player's reasoning but what each believes about the others'.
Common knowledge of rationality
A fact is common knowledge among a group when everyone knows it, everyone knows that everyone knows it, everyone knows that everyone knows that everyone knows it, and so on without end. The standard analysis assumes not only that players are rational but that their rationality is common knowledge: you are rational, I know it, you know that I know it, and so forth ad infinitum. This is a strong assumption — and later modules test what happens when it fails.
Three honesties before we go on. First, the numbers are illustrative; the real fee war is fought over many prices, played month after month rather than once, and between firms of very unequal size — M-Pesa carries the lion's share of Kenya's mobile-money value, well over ninety percent by most counts, so the true game is asymmetric. Second, repetition changes everything: firms that meet again and again, like traders who share a route every day, can sustain restraint through the threat of future punishment, which a one-shot picture hides. Third, the payoffs are not handed down by nature. When the Central Bank of Kenya pushed providers to waive fees on small transfers in early 2020 to keep cash out of pandemic-era hands, it reached into the grid and rewrote the numbers. We lean on all three points later; for now, keep the simplest version fixed so its logic is unmistakable.
From messy reality to a well-posed game
- Players — who are the decision-makers whose choices matter? Decide deliberately who is inside the game and who is part of the fixed environment.
- Actions and strategies — what can each player do? If there is timing, write the full contingent plan, not just the one-off move.
- Outcomes — what results from each combination of choices?
- Payoffs — how does each player rank the outcomes once everything they care about, not just cash, is folded in?
- Information and timing — who knows what, and who moves when? (We keep this simple for now, then relax it in later modules.)
The hardest step in applied game theory is usually not the algebra but this modelling — turning a tangled real situation into a clean object you can analyse. Apply the recipe to the matatu on a SACCO's route. The players are the entrant and the incumbent SACCO. The entrant's actions are to Enter or Stay out; the SACCO's are to Accommodate or Fight. The outcomes are the four combinations, and the payoffs must capture that a shared route is worth something to the entrant but less to the SACCO than a monopoly, while a fare war hurts both. Tabulate that and you get the bimatrix below. Two ideas we have only glimpsed will occupy the coming modules. The first is equilibrium — a combination of strategies at which no player, taking the others' choices as fixed, would want to move. The second is the difference between a pure strategy, which commits to one action, and a mixed strategy, which randomises over actions with set probabilities. Mixing sounds exotic, but it is exactly what a SACCO inspector does when they audit conductors at random so that none can predict, and beat, the inspection. Those tools turn the descriptions in this module into predictions.
Incumbent SACCOAccommodate Fight+-------------+-------------+Matatu Enter | 3 , 6 | −2 , 2 |+-------------+-------------+Matatu Stay out | 0 , 10 | 0 , 10 |+-------------+-------------+
Exercises
Exercise
Use the matatu entry bimatrix above. (a) Name the players and, for each, list its actions; explain the difference between an action and a strategy in this situation. (b) Work out each player's best response to every choice of the other. (c) Predict the outcome, and explain why the SACCO's threat to Fight does not, on these payoffs, deter entry.
Exercise
Return to the M-Pesa vs Airtel fees game. (a) Show, by summing payoffs cell by cell, whether the game is zero-sum. (b) Now suppose the Central Bank of Kenya mandates full interoperability, so a provider that cuts its fee can no longer capture much of the rival's base, while a provider that holds its fee high keeps most of its customers. Concretely, the temptation to undercut falls and being undercut hurts less: the cell where you cut while the rival holds now pays the cutter 9 (was 14), and the cell where you hold while the rival cuts now pays you 6 (was 2). The (High, High) and (Low, Low) cells are unchanged at (10, 10) and (5, 5). Write the new bimatrix as ordered pairs, determine whether either firm now has a dominant strategy, and state the predicted outcome. What has the regulator done to the game?