23506 Chap.6 Mixed-Strategy Applications: Tennis Serve, Entry and Detective Games
Mixed-Strategy Applications: Tennis Serve, Entry and Detective Games
Mixed strategies become concrete when predictability itself is costly. In the two-firm entry game, each firm can enter a market that generates revenue 12 divided by the number of entrants and costs 10. Sole entry earns 2, joint entry loses 4 and staying out gives zero. Each firm enters with probability 1/3 to make the rival indifferent. Independent mixing then gives expected entrants 2/3 and yields a complete distribution over zero, one or two entrants.
Tennis service games apply the same cross-player logic. The server’s direction probability makes the receiver indifferent between positioning choices; the receiver’s coverage probability makes the server indifferent between serve directions. A symmetric first-serve table can yield equal mixing, while altered second-serve success rates or an asymmetric table change equilibrium frequencies. A technically strong direction may be used less often because predictability lets the receiver exploit it.
Detective and auditing games weight targets or violations by their strategic value. With positive target values V and W, symbolic mixes such as V/(V+W) and W/(V+W) must be attached to named players and targets, then tested under symmetric values where both reduce to one-half. Inspection probability deters misconduct; misconduct probability makes costly inspection worthwhile. Before solving an interior mix, check boundary regimes: if entry, attack or violation is never profitable, a pure equilibrium replaces randomisation.
What this chapter covers
- 01Mixed entry and congestion
- 02Outcome probabilities and expected entrants
- 03First- and second-serve tennis
- 04Asymmetric serve probabilities
- 05Detective target values
- 06Auditing, deterrence and comparative statics
Two-firm mixed entry
- 2Calculate entry payoff when the rival stays out and enters.
- 2Let p be the rival’s entry probability and write expected entry payoff.
- 2Set entry payoff equal to the zero stay-out payoff.
- 2Check p lies inside [0,1] and state the symmetric profile.
- 2Calculate expected entrants and verify through event probabilities.
Key terms
- Congestion
- A payoff reduction when more players select the same activity, as in shared entry revenue.
- Strategic unpredictability
- Randomisation that prevents an opponent from exploiting a predictable action.
- Outcome distribution
- Probabilities of realised action profiles implied by mixed strategies.
- Target value
- The payoff consequence attached to protecting, attacking or discovering a particular location.
- Audit probability
- A strategic inspection frequency chosen to affect compliance incentives.
- Comparative statics
- Analysis of how equilibrium choices change when a model parameter changes.
Mixed-Strategy Applications: Tennis Serve, Entry and Detective Games FAQ
Why is entry probability 1/3?
It makes expected entry payoff 2−6p equal the zero payoff from staying out.
What is the expected number of entrants?
Two firms each enter with probability 1/3, so linearity of expectation gives 2/3.
Do event probabilities need independence?
The standard mixed-Nash calculation uses independent private randomisation, allowing profile probabilities to multiply.
Whose tennis probability makes the receiver indifferent?
The server’s direction probability; the receiver’s probability makes the server indifferent.
Does equal-looking action naming imply equal mixing?
No. Check the actual success payoffs and indifference equations.
Why might a stronger serve be used less?
Frequent use makes receiver coverage more attractive, so equilibrium balances technical quality against predictability.
How do target values affect a detective mix?
Higher value changes the opposing player’s incentive, so re-solve the symbolic fractions rather than assuming one-half.
What validates V/(V+W)?
Positive values keep the fraction feasible; symmetric V=W should reduce the relevant probabilities to one-half.
Can auditing have a pure equilibrium?
Yes. If violation is never profitable or inspection is always optimal, an interior mix may disappear.
What should comparative statics explain?
State both the direction of the probability change and whose indifference the change maintains.
How do I know whether an applied mixed equilibrium is plausible?
Check pure boundaries before solving. In entry, ask whether sole entry can cover cost and whether joint entry remains profitable; an interior probability belongs only between pure Enter and pure Stay-out regimes. In tennis, identify which direction each receiver position covers and verify that a pure serve is exploitable. In detective or audit problems, ensure target values, penalties and costs make both actions potentially optimal. Next solve cross-player indifference with explicit labels: the server frequency balances receiver positions, the receiver frequency balances serve directions, protection balances targets and inspection balances compliance. Confirm every probability lies in [0,1]. Translate strategies into event probabilities and verify they sum to one; expected entrant count should agree with the sum of individual entry probabilities. Test symmetric parameter values where applicable, and explain comparative statics through the opponent’s maintained indifference. The equilibrium is plausible when it removes predictable exploitation and every supported action yields the same maximum expected payoff, not merely when the fractions look balanced.
Exam move
For each application, strip away the story and write a 2×2 action/payoff table. Identify the exploitation risk—congestion, predictable direction, exposed target or undetected violation—then run the same cross-player indifference routine. For entry, compute all event probabilities and expected entrant count as redundant checks. For tennis, keep server and receiver probabilities separately labelled and compare first- and second-serve matrices rather than their names. For detective or auditing models, preserve symbolic target values until the end, check positivity and test the symmetric case. Finish by explaining why equilibrium randomisation is useful in that application, not merely reporting a fraction.
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