Venture Capital puzzles, solved step by step

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.

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.

Outside investors should cut the chance the company is good from 50% to 12.5%. Take 200 seed companies: 100 good and 100 bad. The fund backs 90 good and 30 bad ones, and passes on 10 good and 70 bad. A pass therefore comes from a good company 10 times out of 80. An insider investing, by contrast, lifts the chance to 75%.

Why does one fund's decision carry so much information?

Imagine a restaurant owner who eats at her own restaurant nearly every day, and then one week stops. Regulars notice, because she knows the kitchen better than anyone. An existing investor has a board seat, the monthly numbers and a view of the team, so its choice to follow on or not is a signal from the best-informed party in the room. This is the signalling riskThe damage to a startup's fundraising when a well-informed existing investor declines to invest again, because outsiders read the decision as bad news. founders weigh when they take seed money from a large multi-stage fund.

How do you turn the percentages into the answer without a formula?

Use counts, not probabilities; they are harder to get wrong under pressure. Start with 200 companies because it makes every branch a whole number. Of 200, the fund passes on 10 good and 70 bad companies, so a pass is compatible with a good company only 10 times in 80. The same tree answers the mirror question for free: of the 120 companies it backs, 90 are good, so an insider investing lifts the chance from 50% to 75%.

When is the insider's pass weaker evidence than this?

Say the limit before the interviewer does. The numbers assume the fund passes only for reasons of quality. A fund that has run out of reserves, or whose cheque size no longer fits the round, passes for reasons unrelated to the company, and the fall from 50% should then be much smaller. That is why a good outside investor asks why the insider passed before reading the pass as a verdict, and why founders prefer insiders who state their follow-on policy in advance.

There are two equilibria: both fund, worth Rs 26 crore each, and both refuse, worth Rs 9 crore each. If the other funds, your best reply is to fund, Rs 26 crore against Rs 9 crore. If the other refuses, your best reply is to refuse, Rs 9 crore against minus Rs 1 crore. Both funding is better for both, but it is only stable if each trusts the other, so bridges fail on coordination rather than on the economics.

How do you find the equilibria in a two-by-two game?

Two friends agree to meet for a film, and each will only buy a ticket if the other turns up; alone, the ticket is wasted. Both going is best, but both staying home is also a stable outcome, because neither wants to be the one sitting alone. An equilibrium is a pair of choices where each player's choice is the best reply to the other's, so neither gains by switching alone. Work it from each player's seat: fix the other's move, compare your two payoffs, and mark your best reply. Cells where both players' best replies meet are the equilibria.

Where do the payoffs come from?

Each payoff is the value of a 30% stake less the cash put in. Both fund: 30% of Rs 120 crore is Rs 36 crore, less Rs 10 crore, Rs 26 crore. One funds: 30% of Rs 30 crore is Rs 9 crore, so the funder ends at minus Rs 1 crore and the refuser at Rs 9 crore. Neither funds: Rs 9 crore each. Funding together creates Rs 34 crore of value for the two of them, but neither will move first if it fears being left as the lone funder. This is the structure economists call a stag huntA coordination game with two stable outcomes: a better one that needs both players to cooperate, and a safer one each can reach alone..

How do investors get to the good equilibrium in practice?

Remove the fear of funding alone. Bridges are usually written so that no money moves unless the full Rs 20 crore is committed, which deletes the minus Rs 1 crore outcome and leaves funding as the better choice whatever the other does. A lead investor committing first, publicly, does the same job. The limitation of the model is that it fixes both stakes at 30%; in practice the bridge would buy new shares, which raises the reward to funding and makes coordination easier, and a refusing investor may also face penalties such as losing its preferences.

About Rs 120 crore, the second-highest value. As the price rises, Fund A drops out at Rs 100 crore and Fund B at Rs 120 crore. Fund C, which would pay up to Rs 150 crore, wins the moment it bids one step above Rs 120 crore and has no reason to go further. The founder cannot capture C's top value because no one else is willing to push the price there, and C never says what its number is.

Why does the price stop at the runner-up's value?

At a house auction, the buyer who would pay Rs 2 crore often gets the house for Rs 1.6 crore, because the last rival stopped there. In an open ascending auction the winner pays roughly what the second-highest bidder was willing to pay, because the bidding ends the moment the second bidder quits. Fund C's private valuation decides who wins; Fund B's valuation decides the price. The Rs 30 crore between them is the winner's surplus, and it stays with Fund C.

Could the founder run the process differently and get closer to Rs 150 crore?

A sealed bid, where each fund names one price and the highest wins at its own bid, looks as if it would extract more. It usually does not, because funds shade their sealed bids below their true value; under textbook assumptions the expected price ends up about the same as in the open auction. What actually moves the price is more serious bidders: a fourth fund valuing the deal at Rs 140 crore would push the clearing price to Rs 140 crore. That is why bankers and founders work so hard to keep several funds in the process to the end.

Say the limits. Venture rounds are not pure price auctions: founders often take a lower valuation from a fund they want on the board, and term sheets differ on preferences, board seats and option pools, so the headline number is only part of the price. The bid step matters too; the price is Rs 120 crore plus one increment, which in practice is a negotiation rather than a fixed step.

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.

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.

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.

On these numbers, no: waiting is worth Rs 79 crore on average against a certain Rs 80 crore. Weight each outcome by its chance: half of 110 is 55, three tenths of 80 is 24, and the no-deal branch adds nothing. Waiting loses a crore of expected value and adds a one in five chance of having no round at all, which for a company that needs the money can be far worse than zero.

How do you compare a sure thing with a gamble?

A shop offers you Rs 800 for your old phone today, or you can try an online sale that might fetch Rs 1,100, might fetch Rs 800, or might not sell at all. Put a number on the gamble by weighting each outcome by its chance and adding, then compare that expected value with the sure offer. Here waiting gives 0.5 x 110 + 0.3 x 80 + 0.2 x 0, which is Rs 79 crore of pre-money on average, against Rs 80 crore she can sign today.

Why is the expected value not the whole answer?

Because the outcomes are not equally painful. A one crore gap in expected value understates the case against waiting, because the 20% branch is not a lower valuation but no round at all. A company with six months of cash that misses a round may have to cut staff or raise on much worse terms later. Even if waiting were worth slightly more on average, a founder who cannot survive the bad branch should still sign. Turn it round to show the margin: holding the no-deal chance at 20%, waiting only beats signing if the chance of Rs 110 crore rises above 53.3%; holding the 50% upside, the no-deal chance must fall below 18.75%.

What would you ask before trusting the probabilities?

Where the 50% comes from. Founders tend to overrate the chance of a better offer because the meetings that went well are the ones they remember. Ask how many investors are past a partner meeting, and whether the current lead would return at the same price after being turned down, which this puzzle quietly assumes. A pre-money figure is also not the founder's wealth: the stake she keeps depends on how much she raises as well.