Designing Delegation

An Experimental Examination of Veto Bargaining

UCL

June, 2026

Brandon Williams

Alistair Wilson

Richard Van Weelden

Setup

  • Tend to get inefficiency when we consider interactions between uninformed and informed parties:
    • Sender-receiver
    • Market for lemons
    • Bargaining
  • In some of these settings, delegating decision power to the informed party:
    • Potential for efficient solution
    • But share of surplus created is smaller for the party ceding power 

Motivation

Constrained Delegation allows for an intermediate outcome: 

  • The uninformed party delegates some of their decision power to the informed party
  • But imposes some constraints on the choices the informed party can make
    • For example, minimum and maximum limits

Veto bargaining provides an economically important setting to examine this:

  • Veto represents an outside option
  • Theory identifies constrained delegation as optimal mechanism (Kartik, Kleiner, and Van Weelden, 2021)
  • Simple setting for a clear experimental assessment!

Setup

Consider the following bargaining context:

  1. Nature determines the informed party's preference
  2. An uninformed principal makes a proposal
  3. An informed agent responds, comparing the offer it to her outside option

Findings

  • Comparing Constrained Delegation to Take-it-or-Leave it offers we find:
    • Delegation substantially increases efficiency
    • Informed party captures lion's share
    • Proposers fail to optimize, providing more latitude/discretion
    • Preplay communication improves both mechanisms, though Delegation does worse overall
  • Broadly the theory organizes the results well

Literature

Models of similar bargaining forms have been proposed in theory...

  • Under complete information - Romer and Rosenthal (1978)   
  • Under cheap talk negotiation - Matthews (1989)
  • With valuable expertise involved - Holmström (1977, 1984)
  • More recently: Kartik, Kleiner, Van Weelden (2021)

 

And used in experiments...

  • Early bargaining - e.g. Roth and Murnighan (1980, 1982)
  • Vetoes within committees - Kagel, Sung, and Winter (2010)
  • Multiple rounds of bargaining - Nunnari (2021)

 

More recently: Kartik, Kleiner, Van Weelden (2021)

Take a mechanism design approach to show when Proposer:

  • Fully delegates
  • Makes a take-it-or-leave-it offer
  • Offers a limited set of offers

Theory

Theory

Vetoer

0

1

Proposer

\( \theta \)

  • Proposer has:
    • Increasing payoff \( \pi (z) \) over outcome \( z \)
      • \(\pi(z)\) linearly increasing in experiment
    • Makes an offer \(X\subset\left[0,1\right]\) to the vetoer
  • Veteor has:
    • Private information: ideal point \(\theta \) with \( \theta \sim F(\theta) \), preference with convex loss in \(\left|z-\theta\right|\) 
    • Veto point given by \(0\), chooses \(z\in \left\{0\right\}\cup X \)

Theory

0

1

Proposer

Vetoer

\( \theta \)

  • Vetoer's realized preference can be...
    • more aligned with Proposer (\(\theta\) closer to 1)

Theory

0

1

Proposer

Vetoer

\( \theta \)

  • Vetoer's realized preference can be...
    • more aligned with Proposer (\(\theta\) closer to 1)
    • or less aligned  (\(\theta\) closer to 0)

Theory

0

1

Proposer

Vetoer

?

Key experimental/theoretical manipulation for Proposer:

  1. Take it or leave it offer \(X=\left\{ x\right\}\)
  2. Delegation of multiple choices to vetoer \(X\subset\left[0,1\right]\)
    • Interval optimal choice, so \(X=\left[\underline{x},\overline{x}\right]\)

Theory: Take it or Leave It

0

1

Proposer

Offer \( x \)

\( \frac{ x}{2} \)

These types veto

These types take offer

\( z = 0 \)

\( z = x \)

Theory: Take it or Leave It

0

1

Proposer

Offer \( x \)

\( \frac{ x}{2} \)

Suboptimal:

inefficient as \(\theta\)

preferred by both

Breakdown:

inefficient as \( \theta\) preferred by both

Theory: TIOLI equilibrium

  • Marginal increase in offer \(x\) leads to:
    • Marginal gain of \(\pi'(x)\cdot(1-F(\tfrac{x}{2}))\)
    • Marginal loss of \(\left(\pi(x)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{x}{2})\)

 

0

1

\(\tfrac{x}{2}\)

\(z=0\)

\(z=x\)

\(x\)

\(f(\theta)\)

Theory: TIOLI equilibrium

0

1

\(\tfrac{x}{2}\)

  • Marginal increase in offer \(x\) leads to:
    • Marginal gain of \((1-F(\tfrac{x}{2}))\)
    • Marginal loss of \(\tfrac{x}{2}\cdot f(\tfrac{x}{2})\)

\(z=0\)

\(z=x\)

\(x\)

\(f(\theta)\)

\(f(\tfrac{x}{2})\)

Theory: Delegation

0

\( \theta \)

1

Vetoer ideal

Proposer

  • Proposer delegates a set of options \(X\)
  • Vetoer chooses \(z\in\left\{0\right\}\cup X\)
    • either the veto threat point (here \(0\))
    • or some offer in delegation set

Theory: Delegation

0

1

\( \frac{ x}{2} \)

These types veto

These types

choose minimum offer

\( z = 0 \)

\( z = x \)

Offer \( [x,1] \)

\( x\)

These types

choose their ideal point

\( z = \theta \)

Theory: Delegation

0

1

\( \frac{ x}{2} \)

Offer \( [x,1] \)

\(x \)

No suboptimal deals:

The delegation mechanism ensures

these options are available

 

Breakdown:

inefficient as \( \theta > 0 \)

Theory: Delegation Equilibrium

0

1

\(\tfrac{x}{2}\)

\(x\)

\(f(\theta)\)

  • Marginal increase in lower offer \(x\) leads to:
    • Marginal gain of \((F(x)-F(\tfrac{x}{2}))\)
    • Marginal loss of \(\tfrac{x}{2}\cdot f(\tfrac{x}{2})\)

Offer \( [x,1] \)

Theory: Delegation Equilibrium

0

1

\(\tfrac{x}{2}\)

\(x\)

\(f(\theta)\)

  • With increasing density we get a corner solution \(X=\left\{1\right\}\):
    • coincides with TIOLI solution, \(x^\star=1\)

Offer \( [x,1] \)

Theory: Delegation Equilibrium

  • With decreasing density we get the other corner solution:
    • Full delegation, with \(X^\star=\Theta\)
    • Results in first-best for responder with \(z^\star=\theta\)

0

1

\(\tfrac{x}{2}\)

\(x\)

\(f(\theta)\)

Offer \( [x,1] \)

Theory: Summary

  • Delegation should substantially reduce inefficiency
    • Direct Effect: Remove suboptimal deals by providing choice to informed party
    • Indirect Effect: Changing the Proposers' optimization margin leads to more latitude on lower limit, decreasing breakdown
  • Delegation is valuable tool for Proposer under veto threat
  • Proposers use of delegation shifts with expected alignment with Responder:
    • Increasing \(f(\theta)\): No delegation
    • Single peaked \(f(\theta)\): constrained delegation
    • Decreasing \(f(\theta)\): Full delegation

Theory: Environment Characteristics

  • Delegation here is a compromise over veto threat:
    • Proposer has state-independent preferences
    • Tension here is over failure to agree
  • This is not Delegation over information a la Holmstrom (1984)
    • Expert there has valuable information for Proposer's state-dependent preference
    • Tension is over expert's preference misalignment

Theory: Hypotheses

  1. Delegation increases efficiency through:
    • Reduced breakdown
    • Reduced suboptimal deals
  2. Delegation is a valuable tool to Proposers
  3. Delegation moves with expected alignment:
    • Full delegation with decreasing density
    • No compromise with increasing density
    • Constrained with single-peaked density
  4. Cheap talk is:
    • Partially Informative in TIOLI (Matthews QJE 1989)
    • Uninformative in Delegation (Kartik et al AER 2021)

Experiment

Experimental Design

  • Construct environment that directly follows 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

  • Construct environment that directly follows 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: Framing

Proposer

Seller

Vetoer

Buyer

State

Ideal Demand

Offer

Widgets

Delegation

Widget Menu

Types

Urn Draws

Delegation treatment:

offer a range

Take it or leave it:

single offer

Decreasing

probability

Inverse-U shaped

Increasing probability

Experimental Design

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

 

 

 

 

 

No Chat Chat
Take-it-or-leave-it N=66 N=60
Delegation N=64 N=66

Experimental Design

Collect other behavioral variables:                                                 

  • Robot Responder: remove strategic uncertainty/other regarding
  • Pure incentives: present mechanism choice as lottery (remove mechanism complexity)
  • Pure allocation: remove uncertainty and mechanism

Addendum collection (70 more subjects in NoChat):

  • No veto: Remove responder's veto threat
  • Expected Value: remove risk and other regarding
  • CF Mechanism Value: Elicit WTP for the Delegation/TIOLI

Predictions for the Experiment from Theory

Assuming fixed distribution of risk preferences:

  • Within mechanism \(\Phi\) proposer's outcomes ranked: \[ \Phi_{\text{High}} \succ \Phi_{\text{Middle}} \succ\Phi_{\text{Low}} \]
  • Across mechanism for each distribution \(k\) both sender and proposer outcomes ranked:\[\text{DELEG}_k \succ \text{TIOLI}_k\]
  • Communication\(^\star\) before the game:
    • Partial improvements in TIOLI: \[\text{TIOLI}^\star_k \succ \text{TIOLI}_k\]
    • No effect in Delegation: \[\text{DELEG}^\star_k \sim \text{DELEG}_k\]

Results

Low

Middle

High

Sanity check: Proposers respond to alignment

Sanity check: Proposers respond to alignment

Theory expectation: \( \text{TIOLI}_{\text{High}} \succ \text{TIOLI}_{\text{Middle}} \succ \text{TIOLI}_{\text{Low}} \)  ✅

Low

Middle

High

Minimal offer in Delegation interval:

Sanity check: Proposers respond to alignment

Sanity check: Proposers respond to alignment

Theory expectation: \( \text{DELEG}_{\text{High}} \succ \text{DELEG}_{\text{Middle}} \succeq \text{DELEG}_{\text{Low}} \)  ✅-ish

Eqbm Effect: More Latitude with Delegation

Low

Middle

High

Theory expectation:   \(\forall k, \text{DELEG}_k \succ \text{TIOLI}_k \)  ✅

Do Proposers use full interval:

  • Proposers mostly include all possible options in \(\Theta\) above minimum offer \(\underline{x}\)
    • Some exceptions in Low
  • More reluctance to offer options above \(\max(\Theta)\)

Model of Responder Behavior

  • Large majority (~90%) of Responder decisions are selfish/rational
  • The remaining 10% are inefficient, but tend to choose more generously for Proposers
  • Estimate a model of Responder behavior
    • we will later use this model to integrate out the exogenous shock \(\theta\) (not realized until after Proposer decision)

Model of Buyer Behavior

  • Fehr-Schmidt parameterized logit model for Resp. behavior:
    \(U_R(x_R,x_P)=\lambda \left( x_R - \alpha\cdot \max(x_P-x_R,0) - \beta\cdot \max(x_R- x_P ,0) \right)\)

Parameters:

  • \(\lambda\): Response to incentives (Noisier as \(\downarrow 0\))
  • \(\alpha\) : Disadvantageous inequality aversion
  • \(\beta\) : Advantageous inequality aversion
Param TIOLI DELEG p-Val
Scale0.320.53<0.001
Disadv. Ineq 0.12 0.30 0.04
Adv. Ineq 0.14 0.08 0.45

Using model allows us to integrate out exogenous shock \(\theta\) to give expected Proposer/Responder outcomes

Efficiency of Outcomes

Take it or Leave It

Delegation

Delegation decreases inefficiencies

Take it or Leave It

Delegation

Delegation decreases inefficiencies

Rounds with Pareto efficient outcome

UrnTIOLIDELEGp-value
Low45%83%<0.001
Middle53%87%<0.001
High57%83%<0.001
p-value:0.070.60

Result 1:

Conditional Delegation substantially increases efficiency relative to a Take-it-or-leave-it offer. Efficiency increases from both

  • Reduced suboptimality (direct effect)
  • Reduced breakdown (equilibrium effect)

Distribution of Outcome

Who benefits from delegation? 

Low

Decreasing density

Who benefits from delegation? 

Middle

Single-peaked density

Who benefits from delegation? 

High

Increasing density

Who benefits from delegation? 

UrnTIOLIDELEGDiff.p-value
Low$15.62$17.93$2.31<0.001
Middle$20.81$21.97$1.16<0.001
High$25.32$25.51$0.180.217
p-value:<0.001<0.001

Proposer payoffs:

UrnTIOLIDELEGDiff.p-value
Low$24.25$28.11$3.86<0.001
Middle$24.03$27.80$3.77<0.001
High$22.74$26.54$3.81<0.001
p-value:<0.001<0.001

Responder payoffs:

 ✅

 ✅

 ✅

 ✅

 ✅

 ❌

 ✅

 ✅

 ✅

 ✅

Who benefits from delegation? 

UrnTIOLIDELEGDiff.
Low-14.2%-2%12.7pp
Middle-5.4%-0.1%5.3pp
High-1.8%-1.1%0.7pp

Proposer payoffs:

UrnTIOLIDELEGDiff.
Low-19.2%-6.3%12.9pp
Middle-19.9%-7.3%12.6pp
High-24.2%-11.5%12.7pp

Responder payoffs:

Normalized to full-delegation (Responder optimal outcome):

Result 2:

While Proposers do benefit from Delegation -- moreso when they lack bargaining power -- efficiency gains are primarily captured by the Responder.

What would happen if...

Force Buyers to be selfish maximizers?

Force Proposer to max-out top of interval

What explains behavioral deviations?

Additional experimental treatments:

  1. Replace Responder with a payoff-maximizing Robot player
    • Remove strategic uncertainty and other-regarding concerns
  2. Maintain Robot; remove mechanism framing
    • Removes complexity, pure choice over lotteries
  3. Pure other-regarding decision
    • Remove risk, purely distributive decision
  4. Remove veto threat
    • Assess effect of deliberation in dictator analog
  5. Pay with expectation of decision
    • Remove risk maintain strategic uncertainty
  6. Counterfactual assessment
    • Assess behavior and valuation for Alt. Mechanism with experience of other mechanism

Robot Buyer

  • Same decision screen/interface
  • Same payoffs
  • Repeat for all three urns

Remove mechanism framing

  • Choices as a lottery
  • Each lottery option corresponds to  delegation decision
    • Do this for all three urns
    • TIOLI and Delegation

Self/Other Distribution Choice

  • Fix state, look at dictator problem for Seller under complete information
  • Repeat this for low/med/high states

What explains behavioral deviations?

Proposer failure with delegation not driven by other regarding behavior, as when we replace the Buyer with a Robot observed choices closely mirror delegation choices

What explains behavioral deviations?

However, when we remove the mechanism framing,  lottery choices indicate movement toward less latitude in delegation

Replace delegation decisions with lottery?

UrnRobotLottery
Low16.4pp10.7pp
Middle6.5pp7.6pp
High3.5pp7.0pp

\(\Delta\)Proposer payoffs:

UrnRobotLottery
Low14.6pp8.2pp
Middle14.1pp7.1pp
High14.0pp6.5pp

\(\Delta\)Resp payoffs:

Replace delegation decisions with lottery?

  • Sellers do better with Delegation when presented as pure lottery in Middle/High.
    • Esp. when more-aligned with the Responder and they have more bargaining power
  • Suggests optimization failure over their use of the mechanism

How is delegation affected by veto?

  • Can remove the veto threat point to examine value of delegation absent the veto threat mechanic
  • Analagous to comparing Ultimatum vs Dictator game:
    • With uncertainty over Responder preference
    • Selfish BR outcome unaffected by Delegation v. TIOLI

Delegation with No Veto

Without veto-threat having Delegation options reduces the Proposer's payoff and increases the Responder's payoff.

  • Delegation is something you don't want as an option when you have power

Is Delegation valued by Proposers???

  1. Gain Experience: Subjects complete 15 rounds under fixed rule 
  2. Remove risk: Asked to make a Proposer decision paid with the average behavior across many Robots buyers trained to mirror prior subjects
  3. Counterfactual: Explain alternative CF rule, make decision paid with average across Robot buyers trained to CF
  4. Assess relative value: Ask about preference over payment in Parts 2 and Parts 3 where we add/subtract money.

(A. TIOLI or B. DELEG)

(A. DELEG or B. TIOLI)

Behavior across Counterfactuals

  • Looking within-subject at:
    • TIOILI: \(x_i\)
    • DELEG: \([\underline{x}_i,\overline{x}_i]\)
LowMiddleHigh
Move upper bound 6+73%83%94%
Move lower bound below67%72%77%
Both directions49%56%74%
  • Movements
    •  \(\underline{x}_i-x_i>0\)
    •  \(\overline{x}_i\geq6\)

Perceived Value of Delegation

Results: Key Points

  • Proposers respond to the expected alignment
    • Offer more latitude than theory point prediction
    • Delegation offers even more latitude
  • Delegation mechanism substantially increases efficiency
    • This is not true without the veto mechanic
  • Efficiency gains not shared equally:
    • Responder gets the lion's share
    • Value to Proposer well ordered by theory
      • This is also true for their expectations over the CF
    • Proposers give away some of their share through optimization failures
      • But this affects distribution rather than efficiency

Pre-play Communication

Theory: Communication

  • For TIOLI, partially informative
    • Two vague messages
    • High/Low
  • For Delegation, no gain predicted when using interval
    • Only useful for No Compromise outcome (High urn)

Results: Chat

Seller: What are you willing to buy?
Buyer: A middle number will work for me
Seller: Okay, which is better, three or four
Buyer: 3
  • TIOLI, \(\theta=3\), Middle urn
  • Offer is \(x=4\)
  • Outcome \(z=4\)

Results: Chat

Buyer: 0-4 or i walk
Buyer: simple
Seller: hehe
Buyer: ok ok 
Seller: 2-4?
Buyer: so what u gonna offer
Seller: hows that bud
Buyer: hows what bud
Seller: 2-4
  • Delegation, \(\theta=3\), Middle urn
  • Offer is \(X=[2, 4]\)
  • Outcome \(z=3\)

Results: Chat

Buyer: i got 1
Seller: hi
Buyer: pls help me out
Seller: ok ok 
Buyer: tyyy
Seller: i got you
Buyer: :)
Seller: :)
  • Delegation, \(\theta=2\), Low urn
  • Offer is \(X=[1, 6]\)
  • Outcome \(z=2\)

Results: Inefficiency (Data with No Comm)

TIOLI

Delegation

Results: Inefficiency (Data with Comm)

Results: Communication

  • Still a lot of work to do with the communication data
    • Cannot integrate out Responder state realization \(\theta\) as conversation before Proposer choice
  • Use LLMs to code the chats:
    • Across treatments Responders reveal their ideal:
      • 76% in TIOLI (no effect by \(\theta\))
      • 52% in Delegation (decreasing in \(\theta\))
    • Lies are to claim lower values of \(\theta\)
      • False claims in Delegation are much lower than TIOLI
    • More breakdown in Delegation with chat when Responders honestly reveal 

Results: Prop. Outcome (relative to full deleg)

Dist. TIOLI Delegation
Low -14.2% -1.5%
Middle -5.4% -0.1%
High -1.8% -1.1%

Without Chat:

Dist. TIOLI Delegation
Low 9.9% 2.3%
Middle -0.8% -0.1%
High 0.7% 2.6%

With Chat:

Chat Summary

  • Our initial behavioral hypothesis was that chat might hurt Proposers in Delegation (relative to No Chat) through appeals to social preferences.
    • No evidence for this
  • There is some weak evidence that delegation hurts 

Conclusion

  • Examine constrained delegation
    • Here in an ideal setting (veto bargaining) 
  • Participants respond to key tensions from the theory
  • Clear efficiency gains from delegation
    • But more of the surplus goes to the responder
  • An optimization failure in using the mechanism leads to over-delegation
    • However, in expectation this error helps increase efficiency!
  • Preplay communication leads to:
    • Much greater efficiency in TIOLI
    • Slight increase in efficiency for Delegation
    • Proposers do worse with Chat & Delegation than Chat& TIOLI

Thank you!  🙏

Questions or Comments?

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