Research map / Multi-agent systems and games / Games and equilibria
Equal Merit Does Not Imply Equality: Discrimination at Equilibrium in a Hiring Market with Symmetric Agents
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Shows that a symmetric hiring-market bargaining game admits subgame perfect equilibria with asymmetric wages driven by credible threats, and that FTRL on discretized strategies can converge to asymmetric Nash equilibria.
Problem
Discrimination in hiring markets is usually attributed to unequal distributions of resources; the paper asks whether equal-merit, resource-symmetric agents can end up with unequal outcomes at equilibrium via strategy choice in wage bargaining.
Approach
Models hiring wage negotiation as a Rubinstein-style bargaining game with symmetric agents and endogenous outside options; shows that consensus about threat strategies yields a range of SPE with asymmetric payoffs, and that discretized FTRL with an l1 regularizer can converge to asymmetric NE.
Key results
- Theorem 1: for tau <= delta^2/(1+delta), a range of wage splits (w1, w2) is supported at SPE, so candidate groups with equal merit can receive different shares of the surplus.
- Proposition 3: discretized FTRL with l_p regularizer (p=2, D=T, M=O(1/sqrt T)) is no-regret against the best continuous strategy in hindsight.
- Example 6 shows two candidates facing the same firm converge to NE with values 1/D and (D-1)/D under different initial strategies, i.e. almost maximally disparate outcomes.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use consensus strategies with credible threats and endogenous outside options | Symmetric hiring-market bargaining game · Two-type candidate hiring market | equilibrium wage/payoff equality: The largest such gap is given by \frac{\delta-\tau}{1-\tau p} | Hurt |
| Set w_1=w_2 at the upper bound | Symmetric hiring-market bargaining game · Firm and candidate payoffs | payoff gap: \frac{\delta-\tau}{1-\tau} | Hurt |
| Use discretized FTRL with D = T, M=O(1/√T), and p=2 | Two-player finite-round bargaining game · G^(n) | external regret | Helped |
| Use ℓ1-regularized FTRL with the stated initial conditions and tie-breaking | Two-player bargaining game · G^(1) | convergence to Nash equilibrium: M > 2D | Helped |
| Use ℓ1-regularized FTRL under the stated initial-condition constraints | Two-player bargaining game · G^(2) | convergence to Nash equilibrium: D>\frac{1}{1-\delta}; M > 2D | Helped |
| Change the responder's initial strategy from w_c^(1) to w_c'^(1) | Discretized FTRL in G^(2) · Two-player bargaining game | equilibrium value: 1/D; (D-1)/D | Mixed |
| Vary the initial conditions | Discretized FTRL in G^(1) · Two-player bargaining game | payoff symmetry: min {w_{r,1}^{(1)},w_{p,1}^{(1)},α_r} \neq 0.5 | Hurt |
| Use the empirical FTRL setup across the tested initial-condition pairings | Discretized FTRL with ℓ1 regularizer · G^(2), empirical initial-condition pairings | convergence to Nash equilibrium: T=300 | Helped |
Surprising findings
- Equal entitlement to the surplus and symmetric agents still yield asymmetric SPE payoffs, with firms able to capture more surplus than candidates via threats.
- Learning dynamics, not only equilibrium existence, can converge to asymmetric equilibria determined by initial conditions and tie-breaking.
Limitations
- Convergence results for the two-round game are only shown for some initial conditions; full analysis of G(n) is left open.
- Model assumes a large stationary market with fixed type distributions and a simplified two-player finite-round game for learning.
- Only l1 regularizer gives the pure-strategy convergence results; other regularizers are left for future work.
Experimental setup
Metrics: payoff split at equilibrium external regret
Builds on: Rubinstein bargaining Follow the Regularized Leader Ultimatum game
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