Sequential Cursed Equilibrium

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Shengwu Li

Harvard

‘Look before you leap’

• Standard game theory posits that players `look before they leap’
• Examples: Voting, asset markets, auctions.

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Savage (1972): In view of the “look before you leap” principle, acts and decisions, like events, are timeless. The person decides “now” once for all; there is nothing for him to wait for, because his one decision provides for all contingencies.

Savage (1972): ... the task implied by making such a decision is not even remotely resembled by human possibility

People often don’t look before they leap

• They make inferences from observed events
• But neglect the implications of hypothetical events
• Lab experiments:
• Voting
• Esponda & Vespa 2014, 2023
• Asset markets
• Ngangoe & Weizsacker 2021, Martinez-Marquina, Niederle, & Vespa 2019
• Auctions
• Levin, Kagel, & Richard 1996, Levin, Peck, & Ivanov 2016, Nagel, Niederle & Vespa 2025
• Treatment effects not explained by known solution concepts.
• E.g. BNE, cursed equilibrium, level-k, ABEE.

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Outline

1. Cursed equilibrium
• A worked example
• Two problems
2. Sequential cursed equilibrium
• Players ‘look after they leap’
• Solves the two problems
3. Some limitations of sequential cursed equilibrium
4. Evidence from lab experiments

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Cursed equilibrium (Eyster and Rabin 2005)

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Example: Zero-sum trading game

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Sonsino, Erev, Gilat, & Shabtai (2002)

Søvik (2009)

Rogers, Palfrey, & Camerer (2009)

Brocas, Carrillo, Wang, & Camerer (2014)

Problem #1: CE players are not cursed about endogenous information

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Problem #1: CE players are not cursed about endogenous information

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Eyster & Rabin: Treating “exogenous” and “endogenous” private information differently not only seems to us intuitively and psychologically wrong, but also creates some highly artificial differences in predictions based on the way a game is formally written down.

Problem #2: CE players do not learn from observed events.

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Eyster & Rabin: [...] observing actions seems likely to induce more strategic sophistication. Hence, players in certain sequential games may be less cursed than they would be in corresponding simultaneous-move games.

The sequential trading game

• Player 1 moves first.
• If 1 declines, the game ends.
• If 1 accepts, then player 2 moves.

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A solution: Sequential Cursed Equilibrium (SCE)

• Recap:
1. CE players are not cursed about endogenous information.
2. CE players do not learn from observed events.
• We can solve these issues by generalizing CE to extensive games.

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At each type infoset, the player understands the marginal distributions of the other players’ actions and of the state nature moves, conditional on that type infoset, but responds as if the other players’ actions are independent of their types infosets.

At each type, the player understands the marginal distributions of the other players’ actions and of the state, conditional on that type, but responds as if the other players’ actions are independent of their types.

As if ‘actions are independent of infosets’.

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Conditioning on an information set

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Cursed-plausible conjectures

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Cursed-plausibility doesn’t always pin down the conjectures.

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Cursed-plausibility does not tie 1’s conjecture to 2’s actual behavior.

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BUT: At any fully mixed strategy profile, the cursed plausible conjectures are pinned down by Bayes’ rule.

Sequential Cursed Equilibrium

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Existence.

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Theorem: Every finite extensive game of perfect recall has a SCE.

Comparing predictions of BNE, CE, and SCE

🤝 indicates there exists an equilibrium with trade.

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SCE treats exogenous and endogenous information the same way.

SCE players learn from observed events.

Theorem: For any finite simultaneous Bayesian game, ICE and SCE predict the same behavior.

Corollary: For any finite two-player simultaneous Bayesian game, CE and SCE predict the same behavior.

Def: Independently cursed equilibrium (ICE). As in CE, but players respond as if opponents’ actions are independent of each other.

SCE players learn from observed events.

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SCE players can make mistakes ex ante and realize ex post.

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Limitations of SCE

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More solution concepts compared

🤝 indicates there exists an equilibrium with trade.

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Experiment #1: Learning from being pivotal

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Experiment #1: Learning from being pivotal

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Experiment #2: Realized vs unrealized prices

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Experiment #2: Realized vs unrealized prices

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Recap: Experiments that test the theory

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• As far as we know, SCE is the only solution concept that makes these predictions.

Conclusion

• SCE treats infosets the same way that CE treats types.
• If we generalize CE in this way, the result is that:
1. Players are cursed about endogenous information.
2. Players learn from observed events.
• Consistent with some experiments on hypothetical thinking.
• Esponda & Vespa 2014
• Ngangoué & Weizsäcker 2021
• Moser 2019
• Nagel, Niederle, & Vespa 2025

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Partial cursedness

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The ideas adapt to continuous-time auctions.

• Common-value auctions, affiliated signals (Milgrom and Weber 1982)
• Players understand the value of the object:
• conditional on own signal (cursed equilibrium).
• conditional on all observed information (clock cursed equilibrium).

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Result (informal): Under clock cursed equilibrium

1. Bids are lower in Dutch auctions compared to 1^st-price.
• as in Levin, Peck, & Ivanov (2016)
2. Bids are higher in silent English auctions compared to 2^nd-price.
3. The winner’s curse is small in English auctions with many bidders.
• as in Levin, Kagel, & Richard (1996)

Contemporaneous work

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Cursed equilibrium predictions depend on labels.

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