Mixed strategies and randomisation

A game you cannot win by choosing a move

Picture the closing minute of an AFCON final. A striker steps up to the penalty that will decide the trophy; the goalkeeper crouches on the line. The striker can aim left or right; the keeper can dive left or right. Suppose you are the striker and you resolve, firmly, to shoot left. A keeper who reads that resolution dives left and saves it, so you switch to right — but if that plan is readable too, the keeper follows you there. Whatever pure move you commit to, an anticipating keeper beats it; whatever the keeper commits to, you exploit it. There is no move either of you can settle on and rationally keep. In the vocabulary of the previous modules, this game has no pure-strategy Nash equilibrium.

Strip the drama away and the skeleton is Matching Pennies: one player wants to match the other's choice, the other wants to mismatch. Let the striker's payoff be +1 when the ball beats the keeper and −1 when the keeper guesses correctly. The contest is zero-sum, so the keeper's payoff is exactly the negative of the striker's. The stylised payoff matrix — the striker's payoff in each cell — looks like this.

                |  Keeper: L  |  Keeper: R
  Shooter: L    |     −1      |     +1
  Shooter: R    |     +1      |     −1

Run the best responses around the table. If the striker aims left, the keeper wants to dive left; but facing a left-diving keeper the striker prefers right; facing a right-diving keeper the striker's best reply is left again — and so the chase closes into a loop that never settles on a single cell. That circling is the signature of a game with no pure-strategy equilibrium. To find an equilibrium we must enlarge the strategy space: allow each player not to pick a move, but to pick the odds with which they play each move.

Mixed strategies

A mixed strategy is a probability distribution over your available actions. Rather than committing to Left, the striker commits to aiming left with probability p and right with probability 1−p, for some p between 0 and 1; a pure strategy is just the extreme case, p = 0 or p = 1. This is not vagueness but precision about odds — how often, across many identical situations, each action is taken. You evaluate a mixed strategy by its expected payoff, which is linear in the probabilities: the expected payoff of any one of your actions is the probability-weighted average of that action's payoffs across the opponent's moves. If the opponent aims left with probability q and right with 1−q, your Left earns q × (payoff if they go left) + (1−q) × (payoff if they go right), and your Right likewise; your overall payoff then weights the two by p and 1−p. Name both probabilities and the game becomes arithmetic.

The indifference principle

Here is the lever that solves these games. Suppose that in equilibrium you genuinely mix — you place positive probability on both of your actions. That can be rational only if the two actions earn you exactly the same expected payoff. If one paid more, you would shift all your weight onto it and stop mixing. So a player who mixes must be indifferent among the actions they mix over. Now read the consequence carefully: whether you are indifferent depends on how often your opponent plays each of their moves. Therefore, in equilibrium, your opponent must be choosing their mix precisely so as to make you indifferent — and you must be choosing yours so as to make them indifferent. Each player's randomisation is tuned to flatten the other's incentives.

Worked example: KRA audits and the honest-or-evade taxpayer

Bring the idea to a problem a revenue authority faces. A taxpayer files a return and privately chooses whether to report honestly or to evade — under-declaring income to keep money that is legally due. The Kenya Revenue Authority cannot inspect every return, and auditing is costly, so it chooses whether to audit this taxpayer or not. Neither side wants to be predictable: a taxpayer certain of an audit would report honestly, and an authority certain a return was clean would not spend an audit on it — the same circling we met at the penalty spot, now with money and law at stake. Fix three parameters in stylised money units: the tax owed, T = 100; the authority's cost of an audit, c = 20; and the penalty added to the recovered tax when evasion is caught, F = 150. The taxpayer (row) reports Honest with probability 1−q or Evades with probability q; the authority (column) Audits with probability p or not with probability 1−p. Each cell lists the taxpayer's payoff first, then the authority's.

                  |   Audit (p)   |  No audit (1−p)
  Honest (1−q)    |  (−100,  80)  |  (−100, 100)
  Evade  (q)      |  (−250, 230)  |  (   0,   0)

Read the cells. An honest taxpayer pays T = 100 whether or not audited, so their payoff is −100 in both honest cells; the authority collects 100, less the audit cost of 20 when it audits (payoff 80) and the full 100 when it does not. An undetected evader pays nothing (payoff 0) and the authority collects nothing (payoff 0). A caught evader repays the tax plus the penalty, T + F = 250, for a payoff of −250, while the authority recovers 250 less its audit cost of 20, a payoff of 230. Check that no cell is a pure equilibrium: from every corner one of the two players strictly prefers to switch, and the best-response chase loops just as it did before.

Step 1 — the taxpayer's indifference fixes the audit probability

For the taxpayer to mix — to be willing to sometimes evade and sometimes not — the two rows must give equal expected payoff against the authority's audit probability p. Reporting Honest yields −100 regardless of p. Evading yields −250 with probability p (caught) and 0 with probability 1−p (undetected), an expected payoff of p × (−250) + (1−p) × 0 = −250p. Set them equal: −250p = −100, so p* = 100 / 250 = 0.4. The authority must audit with probability 0.4. Notice what fixed that number: T and F — the taxpayer's own stakes. In symbols the indifference condition is −T = −p(T+F), giving p* = T / (T+F) = 100 / 250 = 0.4.

Step 2 — the authority's indifference fixes the evasion probability

For the authority to mix — to be willing to sometimes audit and sometimes not — the two columns must give equal expected payoff against the taxpayer's evasion probability q. Auditing yields 80 when the taxpayer is honest (probability 1−q) and 230 when they evade (probability q): expected payoff 80 × (1−q) + 230 × q = 80 + 150q. Not auditing yields 100 against an honest taxpayer and 0 against an evader: expected payoff 100 × (1−q) + 0 × q = 100 − 100q. Set them equal: 80 + 150q = 100 − 100q, so 250q = 20 and q* = 20 / 250 = 0.08. In symbols, T − c + qF = T − Tq gives q* = c / (T+F) = 20 / 250 = 0.08. The taxpayer evades with probability 0.08 — and that number was fixed by the authority's cost and the penalty, not by anything in the taxpayer's own payoffs.

Comparative statics: the penalty paradox

The compact formulas p* = T / (T+F) and q* = c / (T+F) let us ask how the equilibrium moves when a policy lever moves. Take the obvious lever: raise the penalty F. In q* = c / (T+F), a larger F enlarges the denominator, so the equilibrium evasion rate falls. Harsher penalties do reduce evasion — the intuitive result survives. Why, mechanically? A higher F makes catching an evader more lucrative for the authority, so if evasion stayed as common as before, auditing would strictly dominate; to keep the authority willing to sometimes skip an audit, evasion must become rarer.

Now the surprise. Look at the audit rate p* = T / (T+F). A larger F enlarges that denominator too, so the equilibrium audit probability falls as well: raising the penalty lets the authority audit LESS. The reason runs through the taxpayer's indifference — a bigger penalty makes each audit more frightening, so a smaller chance of being audited already suffices to hold the taxpayer at the knife-edge between honesty and evasion. Penalty severity and audit frequency are strategic substitutes. For a revenue authority this is a genuine result, not a paradox once seen clearly: penalties are cheap threats while audits cost c apiece, so a credible, well-publicised penalty regime can deliver the same deterrence with fewer enforcement resources. Push the numbers — raise F from 150 to 300 and p* falls from 0.4 to 100/400 = 0.25 while q* falls from 0.08 to 20/400 = 0.05: audits and evasion decline together.

Three ways to read a mixed strategy

A probability like 0.08 can unsettle people: does a taxpayer really flip a weighted coin? Mixing admits three interpretations, and in applied work more than one is usually alive at once.

1. Deliberate randomisation. A single decision-maker literally uses a random device. The penalty taker mixes their aim so no keeper can read them; the authority runs a randomised audit lottery so no return is safe. When any detectable pattern would be exploited, randomising is not indecision — it is the optimal policy.
2. Population frequencies. The mixing probability is the fraction of a large population choosing each pure action. Read q* = 0.08 as: within this stylised model, about eight in a hundred returns are evasive; and p* = 0.4 as the share of returns the authority audits. No individual randomises — the aggregate does the mixing, and equilibrium is a statement about proportions.
3. Beliefs. The probability is your opponent's belief about what you will do. You may in fact play a single action, determined by private details of your own situation, while your rival — uncertain which type of taxpayer or striker they face — holds beliefs that, in equilibrium, must be consistent and leave them indifferent.

Match them to the cases. For tax and customs the population-frequency reading is the most natural — the authority cares about the rate of evasion across millions of filings, not about any single coin. For a ranger drawing up a patrol roster or a customs chief assigning search teams, the deliberate-randomisation reading bites hardest: a fixed weekly schedule is a gift to the other side. At the penalty spot, deliberate randomisation and beliefs blend — the striker mixes, and the keeper acts on a read of the striker's tendencies.

Exercises

                  |  Patrol N  |  Patrol S
  Strike N        |    −6      |    +10
  Strike S        |    +4      |    −6

                  |  Search Busia  |  Search Quiet
  Cross Busia     |     −4         |     +8
  Cross Quiet     |     +5         |     −4