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Claim: If choice sets $T, M,$ and $A$ are finite, then an assessment $\{\beta^*_{r}, \beta^*_{s}, \mu^*\}$ is a WPBE (weak perfect Bayesian equilibrium) of the two-stage signalling game between receiver $r$ and sender $s$ if and only if it is a SE (sequential equilibrium). Proof: SE $\implies$ WPBE is trivial since SEs are PBEs by construction, and thus are ...


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Let me answer here putting together all the hints in the comments. You've figured out that the equilibrium bid for a type $v$ bidder is $b(v) = \frac{2}{3}v$. The bids are strictly increasing in $v$, so bidder $i$ wins whenever $b(v_i) \geq b(v_j)$ or $v_i \geq v_j$. The expected payoff for a type $v$ bidder is thus $$ (v -b(v))P(b_1 \geq b_2) $$ or $$ (...


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