Look Around And Pick A Strategic Interaction From Your Day ✓ Solved

Look around and pick a strategic interaction from your day-t

Look around and pick a strategic interaction from your day-to-day life (personal, professional, or news). Make sure that it has some information asymmetry. Write down the situation in words. Then draw a game tree (as in modules 9 or 10). Assign made-up pay-offs. And finally solve the game to find out the Perfect Bayes Equilibrium. Show your work step by step.

Paper For Above Instructions

1. Chosen Strategic Interaction (Description)

Interaction: A job market signaling interaction between an applicant and an employer. Nature first determines the applicant's ability: High (H) or Low (L). Only the applicant observes her own ability (information asymmetry). The applicant then chooses whether to acquire a costly credential (education) or not. The employer observes whether the applicant has the credential but not the applicant's true ability, then offers a wage equal to expected productivity conditional on the observed credential. Payoffs depend on wages and education costs. This is the canonical Spence-style signaling game (Spence, 1973) adapted to a sequential game tree setting that permits Perfect Bayesian Equilibrium analysis.

2. Game Tree (textual description)

Nodes and sequence:

  • Nature: Chooses type t ∈ {H, L} with prior P(H)=p, P(L)=1−p. Type is private information to the applicant.
  • Applicant (knowing type t): Chooses Education decision e ∈ {E (education), N (no education)}. Education is observable to the employer.
  • Employer (after seeing e): Forms belief μ = P(t=H | observed e) and offers wage w(e) (a single action: set wage offer).
  • Payoffs realized: Applicant receives wage w(e) minus cost c_t(e); employer receives productivity v_t minus wage w(e).

We represent the tree branches: Nature → Applicant (E/N) → Employer (wage offer w_E or w_N) → payoffs.

3. Numerical Payoffs and Parameters

Set parameters for clarity and tractability:

  • Prior probability of high type: p = 0.6 (example; results do not require exact p unless checking pooling incentives).
  • Productivities: v_H = 10, v_L = 5.
  • Education costs: If applicant chooses E: c_H(E) = 2 for H type, c_L(E) = 6 for L type (so education is cheaper for high-ability). If applicant chooses N: c_H(N)=c_L(N)=0.
  • Employer offers wage equal to expected productivity given belief μ: w(e) = μ·v_H + (1−μ)·v_L.

Applicant payoff = w(e) − c_t(e). Employer payoff = v_t − w(e).

4. Candidate Strategies and Beliefs (format for PBE)

Strategy of applicant: s_A(H) ∈ {E,N} and s_A(L) ∈ {E,N} specifying education decision by type. Strategy of employer: w_E and w_N specifying wage offers after observing E or N. Belief system: μ(E) = P(H | E) and μ(N) = P(H | N) (updated by Bayes' rule on-path; off-path beliefs must be specified to support equilibria).

5. Solve for Separating Equilibrium (candidate)

Proposed separating strategy: H chooses E, L chooses N. Then observations perfectly reveal type, so beliefs are μ(E)=1 and μ(N)=0. Employer best responses:

  • w_E = μ(E)·v_H + (1−μ(E))·v_L = 1·10 + 0·5 = 10.
  • w_N = μ(N)·v_H + (1−μ(N))·v_L = 0·10 + 1·5 = 5.

Applicant payoffs under these wages:

  • High type: If choose E → payoff = w_E − c_H(E) = 10 − 2 = 8. If deviate to N → payoff = w_N − 0 = 5. Hence H prefers E (8 > 5).
  • Low type: If choose N → payoff = w_N − 0 = 5. If deviate to E → payoff = w_E − c_L(E) = 10 − 6 = 4. Hence L prefers N (5 > 4).

Beliefs comply with Bayes' rule on the equilibrium path. Off-path beliefs are not needed because both actions are on path for their types. Therefore, the candidate separating strategy plus wages (w_E = 10, w_N = 5) and beliefs μ(E)=1, μ(N)=0 constitute a Perfect Bayesian Equilibrium (PBE). This is a stable separating PBE because incentive constraints for both types hold.

6. Check for Pooling Equilibria (brief)

Pooling on E: Suppose both types choose E. Employer belief after E is μ(E)=p=0.6, so w_E = 0.6·10 + 0.4·5 = 6 + 2 = 8. Employer belief after observing N (off-path) can be designed to deter deviations. Applicant payoffs:

  • High type: payoff(E) = 8 − 2 = 6; payoff(deviate to N) = w_N − 0. To prevent deviation, set off-path w_N low (e.g., w_N = 5) by off-path belief μ(N)=0 ⇒ payoff(N)=5. Then H prefers E (6 > 5).
  • Low type: payoff(E) = 8 − 6 = 2; payoff(N) = 5. L would prefer to deviate to N (5 > 2). So pooling on E fails unless employer offers even lower w_N (but off-path beliefs must be credible and consistent with Bayes' rule where possible). In short, pooling on E is not supported with the chosen costs because L has strong incentive to deviate.

Pooling on N (no education): If both choose N, employer offers w_N = p·10 + (1−p)·5 = 8; payoffs: H gets 8 − 0 = 8, but H may want to deviate to E if deviate gives wage w_E (off-path) high enough. Given the parameter choices, a pooling equilibrium on N could be supported by off-path beliefs that assign low probability to H after E, deterring H deviation. Feasible off-path beliefs can sometimes sustain pooling, but separating equilibrium above is the natural and robust PBE here.

7. Formal PBE Statement (separating)

Equilibrium strategies:

  • Applicant: s_A(H)=E, s_A(L)=N.
  • Employer: w_E = 10, w_N = 5.

Beliefs: μ(E)=1, μ(N)=0 (derived by Bayes' rule on path). These satisfy sequential rationality: the employer's wages equal conditional expected productivities, and applicants' educational choices maximize their payoffs given wages. Hence this is a Perfect Bayesian Equilibrium.

8. Step-by-step Solution Summary

  1. Specify game tree: Nature → Applicant (E/N) → Employer (wage) → payoffs.
  2. Assign numeric parameters for v_H, v_L, c_H(E), c_L(E), and prior p.
  3. Propose candidate equilibrium (here separating: H→E, L→N).
  4. Compute employer beliefs using Bayes' rule on-path: μ(E)=1, μ(N)=0.
  5. Compute employer best responses (wages) as expected productivities given beliefs.
  6. Calculate applicant payoffs for each type and each possible action to verify incentive compatibility.
  7. Confirm that no one has profitable unilateral deviations and that beliefs are consistent on-path → conclude PBE.

9. Economic Interpretation

This PBE shows how a credential can separate types when the cost of the credential is sufficiently lower for high-ability applicants. Employers rationally interpret the credential as a signal of high ability and pay accordingly. Low-ability applicants are deterred from mimicking because the credential is too costly relative to the wage premium, so signalling is informative (Spence, 1973). This mechanism explains many real-world credentialing incentives in labor markets (see discussions in Akerlof, 1970; Spence, 1973; Rothschild & Stiglitz, 1976).

References

  • Spence, M. (1973). Job Market Signaling. Quarterly Journal of Economics, 87(3), 355–374.
  • Akerlof, G. A. (1970). The Market for "Lemons": Quality Uncertainty and the Market Mechanism. Quarterly Journal of Economics, 84(3), 488–500.
  • Fudenberg, D., & Tirole, J. (1991). Game Theory. MIT Press.
  • Osborne, M. J. (2004). An Introduction to Game Theory. Oxford University Press.
  • Myerson, R. B. (1991). Game Theory: Analysis of Conflict. Harvard University Press.
  • Signaling literature overview: Riley, J. G. (2001). Silver Signals: Twenty-Five Years of Screening and Signaling. Journal of Economic Literature, 39(2), 432–478.
  • Rothschild, M., & Stiglitz, J. (1976). Equilibrium in Competitive Insurance Markets: An Essay on the Economics of Imperfect Information. Quarterly Journal of Economics, 90(4), 629–649.
  • Kreps, D. M., & Sobel, J. (1994). Equilibrium in Games with Sequential Moves. In A. E. Roth (Ed.), Game-Theoretic Models of Bargaining (pp. 170–188). Cambridge University Press.
  • Conitzer, V., & Sandholm, T. (2008). Game Theory: An Introduction. (lecture notes and surveys for applied game theory)
  • Mailath, G. J., & Samuelson, L. (2006). Repeated Games and Reputations. Oxford University Press.

Related Assignments