Two firms are competing to attract a worker. Each firm simultaneously offers a wage . The worker joins the firm offering the higher wage. If both firms offer the same wage, the worker chooses one firm at random. The worker's value is 100, and firm i's payoff from hiring the worker is
100
−
𝑤
𝑖
, where
𝑤
𝑖
is the wage offered by the firm. If a firm fails to hire the worker, its payoff is 0.
is the wage offered by the firm. If a firm fails to hire the worker, its payoff is 0.

What are and in the

Nash equilibrium respectively?

Question

Two firms are competing to attract a worker. Each firm simultaneously offers a wage . The worker joins the firm offering the higher wage. If both firms offer the same wage, the worker chooses one firm at random. The worker's value is 100, and firm i's payoff from hiring the worker is 
100
−
𝑤
𝑖
, where 
𝑤
𝑖
 is the wage offered by the firm. If a firm fails to hire the worker, its payoff is 0.
 is the wage offered by the firm. If a firm fails to hire the worker, its payoff is 0.

What are  and  in the

Nash equilibrium respectively?

BlackTom AI · Accepted Answer

First, I restate the problem setup to ensure clarity: two firms simultaneously choose wages w1 and w2. The worker picks the higher wage; if tied, picks randomly. The worker’s value is 100, and firm i’s payoff from hiring is 100 − wi; if a firm does not hire, its payoff is 0. The question asks for the Nash equilibrium wages w1 and w2. Now, analyze the strategic structure. If the other firm sets wage wj, then a firm’s best response is to undercut slightly (set wi just below wj) and win the worker, obtaining payoff approximately 100 − wi ≈ 100 − wj + ε, which is positive as long as wj < 100. If a firm matches wj, it would tie and the payoff from hiring would be (assuming a random assignment) 0.5(100 − wj). Since 0.5(100 − wj) ≤ 50 and 100 − wj > 0 for wj < 100, undercutting yields a higher payoff whenever wj < 100. This observation implies there is no pure strategy Nash equilibrium: for any pair of wages (w1, w2) with w1 < 100 or w2 < 100, at least one firm can profit by slightly undercutting the other and securing a payoff close to 100 − the rival’s wage. In particular, if both firms set the same wage w < 100, each would gain by deviating to w − ε to win the worker, since 100 − (w − ε) > 0 and typically exceeds the shared payoff under a tie. Therefore, a pure-strategy Nash equilibrium does not exist in this game when wages are chosen from a continuum [0, ∞) with the constraint that hiring payoff is 100 − wi (and 0 if not hired). What remains is the standard result for this kind of homogeneous-competition setup: there is a mixed-strategy Nash equilibrium in which each firm randomizes over a range of wages (a distribution with support on a subset of [0, 100], since offering more than 100 guarantees you get negative payoff if you win). In such a mixed equilibrium, the exact form of the distribution over wages is determined by the indifference conditions (each firm must be indifferent among the wages in the support), but it does not collapse to a single pair of fixed wages like (100, 100). Because the question asks for specific numerical wages w1 and w2 in the Nash equilibrium, there is a mismatch: in the standard continuous-wage model described, the Nash equilibrium is not a pair of pure strategies (numbers) but a mixed strategy with a distribution over wages. If one were to force a pure-strategy answer, it would not be a Nash equilibrium. If the problem intended a mixed-strategy equilibrium, the answer would be described in terms of the distribution, not just two fixed numbers. In summary: there is no pure-strategy Nash equilibrium with concrete w1 and w2 equal to fixed numbers like (100, 100). The Nash equilibrium exists in mixed strategies, with each firm randomizing over a range of wages (up to but not necessarily including 100) in a way that keeps the opponent indifferent across wages in its support. Without the explicit distribution, a single pair of numerical values (e.g., 100 and 100) cannot be the Nash equilibrium.

23506 Strategic Decision Making and Game Theory - Autumn 2025 Exercises

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