Question
A contractor already holds a service contract worth $8,000 in profit if retained. A challenger may file a competing tender: filing always costs the challenger $3,000, and a successful challenge yields $10,000 before that filing fee. Filing fails if the incumbent has mobilized a defense crew; otherwise the challenger takes the contract and the incumbent gets no contract profit.
The incumbent privately knows its crew-mobilization cost: with probability 40% it is $2,000, and with probability 60% it is $4,000. Everyone knows these probabilities, but only the incumbent knows its own type. After learning its cost, the incumbent chooses whether to mobilize; the challenger simultaneously chooses whether to file, without observing the mobilization choice. Mobilization costs are paid even if nobody files. Players maximize expected monetary profit and can randomize independently conditional on their information.
Find the Bayesian Nash equilibrium: the challenger's filing probability and each incumbent type's mobilization probability. Check that neither player, including either private cost type, wants a pure-strategy deviation.
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Hint 1
The incumbent's defense benefit is realized only when a challenge occurs, but its defense cost is paid even if no challenge occurs.
Check your answer
Solution
Work in thousands of dollars. Let \(e\) be the probability of a challenge and \(d_L,d_H\) the low- and high-cost incumbent mobilization probabilities. If mobilization occurs, the incumbent retains its profit of 8 and pays cost \(c\), whether or not entry occurs. If it does not mobilize, it retains 8 only when there is no entry. Thus the expected incremental value of mobilization for type \(c\) is
\[
(8-c)-8(1-e)=8e-c.
\]
The low type \(c_L=2\) defends whenever \(e>1/4\); the high type \(c_H=4\) defends whenever \(e>1/2\), and each is indifferent at its threshold.
The challenger's expected payoff from filing (before mixing) is
\[
10(1-q)-3,\qquad q=0.4d_L+0.6d_H,
\]
where \(q\) is the unconditional chance that the incumbent mobilizes. Not filing pays zero. Indifference requires \(q=0.7\).
At \(e=1/2\), the low-cost incumbent strictly mobilizes, so \(d_L=1\); the high type is indifferent and may mix. Setting \(0.4+0.6d_H=0.7\) yields \(d_H=1/2\). The challenger then receives \(10(0.3)-3=0\) if it files and 0 otherwise, validating its \(e=1/2\) mix. The low-cost type's defense advantage is \(8(1/2)-2=2>0\), so it never prefers to deviate. The high-cost type's advantage is \(8(1/2)-4=0\), allowing its 50% mixture.
These are the only equilibrium probabilities: for \(e<1/4\) both types prefer not to defend, making filing strictly attractive; for \(1/4<e<1/2\) only the low type defends, giving \(q=0.4\) and still making filing attractive; for \(e>1/2\) both defend, giving \(q=1\) and making filing unattractive. At \(e=1/4\), even maximal low-type defense gives \(q\le0.4<0.7\). Consequently only the high-type indifference point \(e=1/2\) can support the entrant's mixing. The positive mobilization cost paid without entry is what makes the defense types' thresholds depend on the filing probability.
Assumptions
The two players choose simultaneously; the incumbent knows its own cost type, the challenger only its prior.
Filing costs 3 thousand dollars whether or not it succeeds, and gives gross 10 thousand only upon success.
Mobilization costs c thousand regardless of entry; the incumbent earns its existing-contract profit 8 thousand only if it keeps the contract.
Mixed choices are independent conditional on type, and both parties are risk-neutral.
Final Answer
Challenger files with probability 1/2; $2,000-cost incumbent always mobilizes; $4,000-cost incumbent mobilizes with probability 1/2. This is the unique Bayesian Nash equilibrium in behavioral strategies.