Venture Capital puzzles, solved step by step

Puzzles: 100
Traced to a firm: 7
Topics: 12
Hard: 30

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012Five partners, ranked A (most senior) to E, must split Rs 100 crore of carry in whole crores. The most senior remaining partner proposes a split; if at least half of the remaining partners, the proposer included, vote yes, it stands. Otherwise the proposer is removed from the pool and the next most senior proposes. Each partner is purely self-interested and votes no if indifferent. What does A propose?Decision and game theoryHardMulti-stage VCFund of funds and LPs
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How much does partner A keep?
Show the worked solution
A proposes A Rs 98 crore, B nothing, C Rs 1 crore, D nothing and E Rs 1 crore. A needs three of five votes, its own and two more. If A were removed, B's winning plan would give C and E nothing, so Rs 1 crore each buys their votes. Working backwards from two partners up gives every step of the chain.
Where do you start a problem like this?
At the end, where the answer is obvious. A chess player thinks about the final position and works back to the move in front of her. Backward induction solves the smallest game first, then uses its answer as each partner's fallback in the next larger game. With only D and E left, D proposes Rs 100 crore for itself; its own vote is one of two, which is half, so it passes. E gets nothing, and everyone further up the table knows it.
How does each proposer buy votes as the table grows?
With three left, C needs two votes. E gets nothing if C is removed, so Rs 1 crore buys E: C 99, D 0, E 1. With four left, B needs two votes and D is the partner who would get nothing in C's plan, so Rs 1 crore buys D: B 99, C 0, D 1, E 0. Each proposer buys exactly the votes it needs from the partners whose fallback is lowest, paying one crore more than that fallback. With five left, A needs three votes; C and E get nothing in B's plan, so A pays them Rs 1 crore each and keeps Rs 98 crore.
Working up from two partners, each proposer keeps everything except one crore more than the fallback for each vote it needs, so with five partners A pays Rs 1 crore each to C and E, the two who would get nothing under B, and keeps Rs 98 crore.
What assumption is doing the work, and what does the puzzle say about real funds?
The tie rule matters. If an indifferent partner voted yes, A could offer C, D and E nothing and keep all Rs 100 crore, because they get nothing anyway. Stating how indifferent players vote is the step that separates a full answer from a lucky one. The real-world point is that self-interest plus a voting rule rewards whoever controls the agenda, which is why carry splits at real firms are set in writing, with vesting, rather than left to a vote among partners.
Where candidates lose it
Candidates try to reason forward from five partners and get lost, or offer an equal split because it feels fair. The question says purely self-interested; fairness is not on the table, and the only way in is from the end.
The second loss is buying the wrong votes. B and D would do well under B's plan, so they are expensive; C and E get nothing there, so they are cheap. Say who is cheap and why.
What the interviewer asks next
What changes if a proposal needs more than half the votes rather than at least half?
With six partners, what does the most senior propose?
If indifferent partners vote yes, what does A keep?
073Five funds bid for the same deal. Each estimates its value with an independent error spread evenly between minus 20% and plus 20%, and bids exactly its estimate. On average, by how much does the winning fund overpay?Decision and game theoryHardGrowth equityMulti-stage VC
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The winner's average overpayment is about:
Show the worked solution
About 13.3%. Every fund's estimate is right on average, but the deal goes to the fund with the highest estimate, and the highest of five errors is not average. For errors spread evenly from minus 20% to plus 20%, the highest of n sits on average (n - 1)/(n + 1) of the way to the top: with five bidders that is 4/6 of 20%, or 13.3%. Winning is itself evidence that you guessed high.
Why does winning tell you that you overestimated?
Ask five friends to guess the weight of a goat at a fair and give the goat to whoever guesses highest; the winner will nearly always have guessed too much. When the prize goes to the highest estimate, the winner is selected for being too optimistic, so even unbiased bidders overpay on average. This is the winner's curseThe tendency for the winner of an auction with uncertain value to have overestimated that value, because winning selects the most optimistic estimate.. No one bid foolishly here; each estimate was fair. The overpayment comes entirely from the selection.
One fund's valuation error is spread flat around zero, but the highest of five errors bunches near the top of the range and averages plus 13.3%, which is how much the winning fund overpays on average.
The relationship
$E[\max \text{ of } n] = \frac{n-1}{n+1} \times 20\% = \frac{4}{6} \times 20\% = 13.3\%$
n number of bidders, 5
20% the widest error any one estimate can have
(n - 1)/(n + 1) how far toward the top the highest of n evenly spread values sits on average
What it says in wordsThe best of five evenly spread errors sits on average two-thirds of the way to the top, so the winner is about 13% too high.
What should a disciplined bidder do about it?
Bid below your estimate, and more so when there are more bidders or more uncertainty. The curse grows with the number of bidders and with the width of the errors: two bidders overpay by 6.7% on average, five by 13.3%, ten by 16.4%. So a fund in a crowded growth round should shade its bid by roughly the expected overestimate, about 12% of its estimate here, and a fund that keeps winning contested deals should ask whether it is better informed or just more optimistic. A simulation of 200,000 auctions in the source file gives 13.3%, matching the formula.
Bidders Winner's average overpayment
1 0.0%
2 6.7%
5 13.3%
10 16.4%
With errors spread evenly up to 20% either way, the winning bid's average overestimate rises with the number of bidders; one bidder has no curse at all.
The limit: the model assumes every fund's estimate is equally noisy and that all bid their raw estimate. Better-informed funds face a smaller curse, and in practice everyone shades, so real overpayment is smaller than the raw figure. The direction of the effect is what to carry into the room.
Where candidates lose it
The fast wrong answer is zero, from the true fact that each estimate is unbiased. The mistake is forgetting that the winner is not a random fund; it is the one with the highest estimate.
The second trap is answering 20%, as if the winner always has the maximum error. The highest of five sits two-thirds of the way up on average, not at the edge; give the (n - 1)/(n + 1) rule and the 13.3%.
What the interviewer asks next
How does the overpayment change with ten bidders?
If one fund has half the error of the others, how should it bid?
Why do proprietary deals, with one bidder, avoid the winner's curse?

n	number of bidders, 5
20%	the widest error any one estimate can have
(n - 1)/(n + 1)	how far toward the top the highest of n evenly spread values sits on average

Bidders	Winner's average overpayment
1	0.0%
2	6.7%
5	13.3%
10	16.4%

Try it first

Where do you start a problem like this?

How does each proposer buy votes as the table grows?

What assumption is doing the work, and what does the puzzle say about real funds?

Where candidates lose it

What the interviewer asks next

Try it first

Why does winning tell you that you overestimated?

What should a disciplined bidder do about it?

Where candidates lose it

What the interviewer asks next