Veto Power, Delegation and Mechanism Design

Brandon Williams

Alistair Wilson

Richard Van Weelden

BEEMA9, Lehigh University

September 12, 2025

Setup

Setup

  • Many "bargaining" contexts exist in which a less-informed party must decide what to offer to a more-informed party, who hold veto power

Setup

  • Many "bargaining" contexts exist in which a less-informed party must decide what to offer to a more-informed party, who hold veto power

Setup

  • Many "bargaining" contexts exist in which a less-informed party must decide what to offer to a more-informed party, who hold veto power

 

 

 

 

 

  • The proposers:
    • Can make simple "take it or leave it" offers
    • Or they can offer a range of options to the informed party, conceding some of their agenda-setting power

Setup

  • Many "bargaining" contexts exist in which a less-informed party must decide what to offer to a more-informed party, who hold veto power

 

 

 

 

 

  • The proposers:
    • Can make simple "take it or leave it" offers
    • Or they can offer a range of options to the informed party, conceding some of their agenda-setting power
  • Theory suggests that the delegation mechanism should capture any alignment friction and increase efficiency
  • Behaviorally, this may not be the case

Theory

Theory

Kartik, Kleiner, Van Weelden (2021)

Theory

0

1

Theory

Proposer

0

1

Theory

Proposer

0

1

  • Proposer has:
    • Increasing payoff \( \pi (z) \) over the outcome \( z \) (for simplicity, we'll assume this is linear)
    • Will make an offer to the vetoer

Theory

\( \theta \)

Vetoer

0

1

Proposer

  • Veteor has:
    • Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)                                    

Theory

Vetoer

0

1

Proposer

\( \theta \)

  • Veteor has:
    • Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)                                  

Theory

???

0

1

\( \theta \)

  • Veteor has:
    • Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)                                  

Theory

???

0

1

  • Veteor has:
    • Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)                  
    • A choice \( z \in \{0,Y\} \), either the veto threat point (here 0) or in the offer

\( \theta \)

Theory

  • Veteor has:
    • Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)                  
    • A choice \( z \in \{0,Y\} \), either the veto threat point (here 0) or in the offer

0

1

Theory: Take it or Leave It

0

\( \theta \)

1

Theory: Take it or Leave It

0

\( \theta \)

1

Vetoer

Proposer

Theory: Take it or Leave It

0

1

Proposer

Offer \( y \)

\( \frac{ y}{2} \)

These \( \theta \)-types veto

These \( \theta \)-types choose offer

\( z = 0 \)

\( z = y \)

Theory: Take it or Leave It

0

1

Proposer

Offer \( y \)

\( \frac{ y}{2} \)

Breakdown:

inefficient as \( \theta > 0 \)

Suboptimal:

inefficient as \( \theta > y \)

preferred by both

Theory: Take it or Leave It

0

1

\( \frac{ y}{2} \)

\(f(\theta)\)

\( y \)

\( z = y \)

\( z = 0 \)

  • Take it or leave it equilibrium depends on the distribution \( F(\theta) \)
  • Increasing the offer \( y \) leads to:
    • Marginal gain of \(\pi'(y)\cdot(1-F(\tfrac{y}{2}))\)
    • Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

Theory: Delegation

0

\( \theta \)

1

Theory: Delegation

0

\( \theta \)

1

Vetoer

Proposer

  • Proposer:
    • Offers set of options \(Y\)
  • Vetoer has:
    • Chooses \(z\in\left\{0\right\}\cup Y\), either the veto threat point (here \(0\)) or some offer in delegation set

Theory: Delegation

0

1

\( \frac{ y}{2} \)

These \( \theta \)-types veto

These \( \theta \)-types

choose minimum offer

\( z = 0 \)

\( z = y \)

Offer \( [y,1] \)

\( y \)

These \( \theta \)-types

choose their preferred

\( z = \theta \)

Theory: Delegation

0

1

\( \frac{ y}{2} \)

Offer \( [y,1] \)

\( y \)

No suboptimality:

The delegation mechanism ensures

these options are available

 

Breakdown:

inefficient as \( \theta > 0 \)

Theory: Delegation

0

1

  • Delegation equilibrium also depends on the distribution \( F(\theta) \)
  • Increasing the minimum offer \( y \) leads to:
    • Marginal gain of \(\pi'(y)\cdot(F(y)-F(\tfrac{y}{2}))\)
    • Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

\(f(\theta)\)

\( \frac{ y}{2} \)

\( y \)

Theory: Delegation

0

1

\(\tfrac{y}{2}\)

\(y\)

\(f(\theta)\)

  • Two special cases:
    • With increasing density we get a corner solution:
      • Coincides with TIOLI solution, \(y^\star=1\)

Theory: Delegation

0

1

\(\tfrac{y}{2}\)

\(y\)

\(f(\theta)\)

  • Two special cases:
    • With increasing density we get a corner solution:
      • Coincides with TIOLI solution, \(y^\star=1\)
    • With decreasing density we get the other corner solution:
      • Full delegation, with \(Y^\star=\Theta\) and \(z^\star=\theta\)

Theory: Summary

  • Delegation should capture a meaningful proportion of alignment failures, and therefore have more efficient outcomes than take it or leave it offers
  • In both cases, proposers should change their offer according to ex-ante alignment (i.e. bargaining power)
  • Theory informs the optimal offer, including when no delegation and full delegation are optimal
  • We test if these predictions hold and assess behavioral deviations (e.g. other regarding behavior, optimization failures, etc.)

Experiment

Experimental Design

  • Constructed environment that models the veto bargaining framework
  • One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

  • Constructed environment that models the veto bargaining framework
  • One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

  • Constructed environment that models the veto bargaining framework
  • One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?

Experimental Design

  • Constructed environment that models the veto bargaining framework
  • One challenge: how to bring this abstract environment to a participants in a way that is easier to understand?
  • To make it tractable:
    • Frame as a Buyers and Sellers problem:
      • State is the Buyer's ideal demand
      • Delegation is choice of menu to offer
    • Discretize the state space to 6 states (draws of a ball) with 3 distributions (urn types):
      • High (increasing)
      • Middle (modal distribution)
      • Low (decreasing)

Delegation treatment:

offer a range

Take it or leave it:

single offer

Experimental Design

  • Within-subject variation:
    • Varying distributions (high, middle, low) for the Buyer           
    • Changing roles: 5 rounds in one role, 5 rounds in the other, and back to first role for 5 more rounds
  • Between subject 2x2

 

 

 

 

 

  • 12 sessions at the Pittsburgh Experimental Economics Laboratory
No Chat Chat
Take-it-or-leave-it N=66 N=60
Delegation N=64 N=66

Experimental Design

  • Within-subject variation:
    • Varying distributions (high, middle, low) for the Buyer           
    • Changing roles: 5 rounds in one role, 5 rounds in the other, and back to first role for 5 more rounds
  • Between subject 2x2
  • 12 sessions at the Pittsburgh Experimental Economics Laboratory
  • Collect other behavioral variables:                                                  
    • Optimizing ability while playing against a robot Buyer
    • Risk-aversion over the same domain
    • Other-regardingness also over the same domain

Results

Results: Seller offers in TIOLI

Low

Middle

High

Results: Seller offers in TIOLI

Low

Middle

High

Results: Seller offers in TIOLI

Results: Seller offers in TIOLI

Results: Seller offers in Delegation

Low

Middle

High

Minimal offer in interval:

Results: Seller offers in Delegation

Low

Middle

High

Minimal offer in interval:

Results: Offer comparison over Mechanisms

Low

Middle

High

Results: Offer comparison over Mechanisms

Low

Middle

High

Results: Inefficiency (Theory)

TIOLI

Delegation

Results: Inefficiency (Theory)

TIOLI

Delegation

Results: Inefficiency (Theory)

TIOLI

Delegation

Results: Inefficiency (Data)

TIOLI

Delegation

Effect of Delegation: Theory

Effect of Delegation: Theory

Effect of Delegation: Empirical Payoffs

Results: Key Points

  • Offers respond to the distribution of the types in a well-ordered manner
    • Offers are more generous than perfectly-optimized predictions
    • Proposers/sellers in delegation are more generous 
  • Delegation mechanism is more efficient than take it or leave it, which is predicted the theory
  • Most of the efficiency gains are captured by the vetoer/buyer, which is not predicted by the theory
  • Best improvements for the proposer/seller is in the decreasing (low) distribution

Other Results

What is the effect of communication?

  • Communication improves bargaining efficiency
    • More in take it or leave it case, where we might expect it
    • But also in delegation
  • Proposers/sellers do worse in the delegation case with chat

Why do we see behavioral deviations from theory?

  • Main reason sellers don't extract more of the delegation gain is optimization failure
    • Lottery choices indicate less delegation
    • Robot choices match behavior in the Delegation game

 

Conclusion

  • Test delegation bargaining with veto power in a lab setting 
  • Proposers respond to key distribution parameters and change their offers accordingly
  • Clear efficiency gains from the delegation mechanism over take-it-or-leave-it offers
    • But more of the surplus goes to the responder
  • Optimization failures in understanding the mechanism action space explain some of the proposer's failure to extract more
    • However, this doesn't lead to inefficiency as they over-delegate

Thank you!

Questions or Comments?

Diagnosing the Failures: Pure Optimization

Diagnosing the Failures: Lotteries

Diagnosing the Failures: Distribution

Results: What else is included in interval?

Results: What else is included in interval?