Auctions for Risk Averse Charities

Joshua Foster at Ivey
Jeffrey Carpenter & Peter Matthews
at Middlebury College

Your favorite non-profit needs $\$$10,000.

Donor Community

50% chance @ 10k

50% chance @ 12k

New Wealthy Donor

10% chance @ 150k

90% chance @ 0

Your favorite non-profit needs $\$$10,000.

Winner-pay Auctions

50% chance @ 10k

50% chance @ 12k

All-pay Auctions

10% chance @ 150k

90% chance @ 0

Valuations

Bids

Revenues

Our focus.

  1. Is there a theoretical/experimental mean-variance tradeoff among fundraising mechanisms?
  2. How might this be a function of participation costs?
  3. What advice can we give on fundraising design?

Experimental Design Overview

Tested 10 mechanisms with 5 sessions each.

10 potential bidders and 10 auctions per session.

Public good benefits imposed on auction revenue.

Phase 1: Earn Money

  • 16 incentivized anagram puzzles.
  • Difficulty exogenously varied.

Phase 2: Bid or Anagram

  1. Participate in an auction.
  2. Attempt another anagram (of known difficulty).

Imposed Mechanisms

First-price Winner-pay
Sealed bid, Dutch

Second-price Winner-pay
Sealed bid, English, Silent

First-price
All-pay
Sealed bid

Second-price All-pay
All-pay Button, "Bucket"

Last-price
All-pay
Sealed bid

Standard Lottery
Raffle

Standard IPV Framework

Expected ex ante bid $\overline{b}_m$ through mechanism $m$:

$\overline{b}_m=\int_{\hat{v}_m}^{\overline{v}}b_m(v)dF_m(v)$

With $n$ bidders, expected revenue is $\mu_m=n\overline{b}_m$.

Bidder's revenue variance:

$\text{Var}(b_m)=\int_{\hat{v}_m}^{\overline{v}}b_m(v)^2dF_m(v)-\overline{b}_m^2$

Variance $\sigma_m$ of mechanism $m$:

$\sigma_m^2=\text{Var}\left(\sum_{i=1}^n b_m^i\right)$

$=\sum_{i=1}^n\text{Var}(b_m^i)+2\sum_{i\leq j}\text{Cov}(b_m^i,b_m^j)$

Endogenous Participation

The $n$ potential bidders face a cost $c$ for participation.

  • Motivation: bid preparation, fairness, etc.
  • Experiment: expected value of an anagram puzzle.
  • Upshot: active bidders is not pre-determined.

There exists a mechanism-specific fixed threshold value $\hat{v}$ that determines participation.

Theoretical Mean-variance Tradeoff by Mechanism, Participation Cost

Theoretical Mean-variance Tradeoff by Mechanism, Participation Cost

Prediction 1: Mechanisms with greater expected revenue also have greater variance.

Theoretical Mean-variance Tradeoff by Mechanism, Participation Cost

Prediction 2: Winner-pay variance will increase with participation cost, while all-pay variance will decrease.

Result 1: Nearly all mechanisms with greater expected revenue also have greater variance.

Participation & Revenue Variance by Puzzle Difficulty (OLS)

Participation & Revenue Variance by Puzzle Difficulty (OLS)

Result 2: An $\uparrow$ participation costs $\Rightarrow$ revenue variance to $\uparrow$ if winner-pay, but $\downarrow$ if all-pay.

Fundraising design: a "balanced" event.