Constrained Delegation

Lessons for Behavioral Mechanism Design

University of Arizona

January 2026

Brandon Williams

Alistair Wilson

Richard Van Weelden

Setup

Tend to get inefficiency when we consider interactions between uninformed and informed parties:
- Sender-receiver
- Bargaining
- Market for lemons
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:

Nature determines the informed party's preference
An uninformed principal makes a proposal
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:
- Constrained delegation does increase efficiency a lot
- However, the informed party still captures almost all of the benefits behaviorally
- Proposers fail to optimize the design problem over delegation, providing more latitude/discretion
- Pre-play communication yields similar gains across mechanisms

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 choice with constraints

Theory

Theory

Proposer

Theory

Proposer

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

Theory

Vetoer

Proposer

\( \theta \)

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

Theory

???

\( \theta \)

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

Theory

???

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
- Preference represented by convex loss from ideal \(\theta\)
- Takes choice \( z \in \{0,y\} \): either the veto threat point (here 0) or the offer \(y\).

\( \theta \)

Theory

Veteor has:
- Ideal point \( \theta \) which is private information, \( \theta \sim F(\theta) \)
- Preference represented by convex loss from ideal \(\theta\)
- Takes choice \( z \in \{0,y\} \): either the veto threat point (here 0) or the offer \(y\).

Theory: Take it or Leave It

Proposer

Offer \( y \)

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

These types veto

These types take offer

\( z = 0 \)

\( z = y \)

Theory: Take it or Leave It

Proposer

Offer \( y \)

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

Suboptimal deals:

inefficient as \( \theta > y \)

preferred by both

Breakdown:

inefficient as \( \theta > 0 \) preferred by both

Theory: TIOLI equilibrium

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \(\pi'(y)\cdot(1-F(\tfrac{y}{2}))\)
- Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

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

\(z=0\)

\(z=y\)

\(y\)

\(f(\theta)\)

Theory: TIOLI equilibrium

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

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \((1-F(\tfrac{y}{2}))\)
- Marginal loss of \(\tfrac{y}{2}\cdot f(\tfrac{y}{2})\)

\(z=0\)

\(z=y\)

\(y\)

\(f(\theta)\)

Theory: Delegation

\( \theta \)

Vetoer ideal

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

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

These types veto

These types

choose minimum offer

\( z = 0 \)

\( z = y \)

Offer \( [y,1] \)

\( y \)

These types

choose their ideal point

\( z = \theta \)

Theory: Delegation

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

Offer \( [y,1] \)

\( y \)

No suboptimal deals:

The delegation mechanism ensures

these options are available

Breakdown:

inefficient as \( \theta > 0 \)

Theory: Delegation Equilibrium

Proposer increasing the minimal offer \(y\) leads to:
- Marginal gain of \(\pi'(y)\cdot(F(y)-F(\tfrac{y}{2}))\)
- Marginal loss of \(\left(\pi(y)-\pi(0)\right)\cdot \tfrac{1}{2}f(\tfrac{y}{2})\)

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

\(y\)

\(f(\theta)\)

Theory: Delegation Equilibrium

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

\(y\)

\(f(\theta)\)

Proposer increasing the offer \(y\) leads to:
- Marginal gain of \((F(y)-F(\tfrac{y}{2}))\)
- Marginal loss of \(\tfrac{y}{2}\cdot f(\tfrac{y}{2})\)

Theory: Delegation Equilibrium

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

\(y\)

\(f(\theta)\)

With increasing density we get a corner solution:
- coincide with TIOLI solution, \(y^\star=1\)

Theory: Delegation Equilibrium

With decreasing density we get the other corner solution:
- Full delegation, with \(Y^\star=\Theta\) and \(z^\star=\theta\)

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

\(y\)

\(f(\theta)\)

Theory: Summary

Delegation should substantially reduce inefficiency
- Remove all the suboptimality by providing choice
- By changing the margins, leads to more latitude in offers, so decreasing breakdown too
Delegation should improve the proposer's payoffs
Proposers offers should shift with the distribution of responders

Theory: Environment Characteristics

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

Theory: Hypotheses

Delegation improves Proposer outcomes over TIOLI
Delegation is:
- Full delegation with decreasing density
- No compromise with increasing density
- Constrained with unimodal density
Proposer provides increased discretion as Proposer's risk aversion increases
Cheap talk is:
- Informative in the TIOLI set-up (Matthews QJE 1989)
- Uninformative with used 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

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 Buyer
- 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

Within-subject variation:
- Varying distributions (high, middle, low) for the Buyer
- Between subject 2 x 2
Collect other behavioral variables (identification through subtraction):
- Robot Buyer: remove strategic uncertainty/other regarding
- Pure incentives: present mechanism choice as lottery (remove mechanism complexity)
- Pure allocation: remove uncertainty and mechanism

Experimental Predictions

Varying distributions for the Buyer ideal provide a sanity check on understanding
- Full Delegation with decreasing density
- No Delegation with increasing density
Conditional Delegation predicted to:
- substantially increase efficiency
- increase Seller payoff (relative to TIOLI)
- discretion increases with Seller risk aversion
Communication before the game:
- Partial improvements in TIOLI
- No effect in Delegation

Results

Low

Middle

High

Sanity check: Sellers respond to alignment

Low

Middle

High

Minimal offer in interval:

Sanity check: Sellers respond to alignment

Low

Middle

High

Minimal offer in interval:

Sanity check: Sellers respond to alignment

Low

Middle

High

Sellers offer more latitude under delegation

Other quick results

Buyers overwhelmingly pick the best option available
- They pick the best number of widgets from the menu, and take the outside option when better
Sellers (mostly) keep offers higher than the minimum
- They mostly refrain from modifying the upper bound

Model of Buyer Behavior

Large majority (~90%) of Buyer decisions are selfish/rational
The remaining 10% are inefficient, but tend to choose more generously for Sellers
Estimate a model of Buyer behavior to integrate out exogenous shocks

Model of Buyer Behavior

Fehr-Schmidt parameterized logit model for Buyer behavior:
\(U_B(x_B,x_S)=\lambda \cdot x_1 + \alpha\cdot \max(x_S-x_B,0) + \beta\cdot \max(x_B- x_S ,0) \)

Here express relative parameters:

Relative effect of disadvantageous inequality: \(\tfrac{\alpha}{\lambda}\)
Relative effect of advantageous inequality:\(\tfrac{\beta}{\lambda}\)

Param	TIOLI	Delegation
Disad. Inequality	-0.08	-0.43
Adv. Inequality	-0.20	-0.17

Using model allows us to integrate out exogenous shock \(\theta\) when forming expectations

Efficiency of Outcomes

Delegation should increase efficiency

Take it or Leave It

Delegation

Delegation should increase efficiency

Take it or Leave It

Delegation

Take it or Leave It is inefficient

Take it or Leave It

Delegation

Inefficiency not eliminated under delegation

Take it or Leave It

Delegation

Delegation does increase efficiency

Take it or Leave It

Delegation

Take it or Leave It

Delegation

Delegation does increase efficiency

Result 1:

Conditional Delegation drastically increases efficiency in our experiments relative to Take-it-or-leave-it offers.

Distribution of Outcome

Who benefits from delegation?

Low

Who benefits from delegation?

Middle

Who benefits from delegation?

High

Result 2:

Efficiency gains from Conditional Delegation yields more gains for Buyer (though this mirrors the theoretical predictions). When Seller has more bargaining power, Conditional Delegation backfires.

Replace Buyers with Selfish Player

Replace Buyers with selfish Robots?

What explains behavioral deviations?

Additional treatments at the end:

Replace Buyer with a payoff-maximizing Robot player
- Remove strategic uncertainty and other-regarding concerns
Maintain Robot, but remove mechanism framing
- Remove complexity, pure choice over lotteries
Pure other-regarding decision
- Remove risk, purely distributive decision

Robot Buyer

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

Remove mechanism framing

Choices are lotteries
Each lottery option corresponds to delegation decision
- Do this for all three urns
- For TIOLI and Delegation

Self/Other Distribution Choice

Fix unknown state, look at dictator problem for Sellers
Repeat this for low/med/high states

What explains behavioral deviations?

Main reason sellers don't extract more of the delegation gain is optimization failure:

Remove player: Robot choices closely match delegation choices

What explains behavioral deviations?

Remove player: Robot choices closely match delegation choices
Remove mechanism complexity: Lottery choices indicate pure preference is for less delegation

Other Choices

Replace delegation decisions with lottery?

Sellers do better with Conditional Delegation when presented as pure lottery
- Do not find any substantive difference for TIOLI
Suggests an optimization failure over the mechanism

Delegation discretion and Risk Pref.?

Low

Middle

High

Results: Key Points

Offers respond to the expected type in a well-ordered manner
- Sellers offer more latitude than theoretical prediction
- Sellers in delegation offer ever more latitude
Delegation mechanism is more efficient than take it or leave it
- Responds to Seller risk preferences, no effect in TIOLI
Most of the "cost" of the mechanism falls on the Seller:
- More of the efficiency gains are captured by the Buyer
- Best improvements for the Seller is when alignment is low
- Data suggest that the optimization failures are a reason for greater latitude in delegation

Pre-play Communication

Theory: Communication

For TIOLI, partially informative
- Two vague messages
- High/Low

For Delegation, no gain over conditional delegation
- 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 \(y=4\)
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 \(Y=[2, 4]\)
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 \(Y=[1, 6]\)
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 Buyer state realization as this is potentially revealed during conversation
We use Gemini 3 to code the chats:
- Across treatments Buyers reveal their ideal:
  - 69% in TIOLI (no effect by \(\theta\))
  - 42% in Delegation (decreasing in \(\theta\))
- Sellers concede 41% of the time to Buyer ideal
  - More haggling by Seller in TIOLI
Still tuning some of the chat coding

Results: Seller Outcome (relative to full deleg)

Dist.	TIOLI	Delegation
Low	-14%	+1%
Middle	-7%	+4%
High	+3%	-7%

Without Chat:

Dist.	TIOLI	Delegation
Low	+20%	+13%
Middle	+13%	0%
High	+8%	+5%

With Chat:

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
- Additional tensions in communication

Thank you! 🙏

Questions or Comments?

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