Venture Capital puzzles, solved step by step
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028A multi-stage fund invests in its seed companies' Series A for 90% of the good ones and 30% of the bad ones. Half of its seed companies are good. It passes on one company's Series A. What should outside investors now believe about that company?Multi-stage VCSeed and early-stage VC
Try it first
After the insider passes, what is the chance the company is a good one?
Show the worked solution
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.
Of 200 seed companies, the insider backs 90 good and 30 bad ones and passes on 10 good and 70 bad ones, so only 12.5% of the companies it passes on are good, against 75% of those it backs. 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%.
The relationship0.5 the share of seed companies that are good, and the share that are bad 0.1 the chance the fund passes on a good company 0.7 the chance the fund passes on a bad company What it says in wordsOf all the passes, the share that come from good companies is the answer.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.
Where candidates lose it
Candidates answer 10% because they see the 90% invest rate for good companies and flip it. That is the chance of a pass given a good company, not the chance of a good company given a pass, the classic confusion of the two conditionals.
The second loss is staying at 50% because it is only one decision. One decision from the best-informed investor moves the odds a long way; draw the tree with 200 companies and show how far.
What the interviewer asks next
- How good would the fund's judgement need to be for a pass to leave the odds at 40% rather than 12.5%?
- What can a founder with a multi-stage seed investor do to reduce signalling risk?
- If a third of all passes are for fund reasons unrelated to quality, what is the chance a passed company is good?
040Two co-investors each own 30% of a company that needs a Rs 20 crore bridge. Each can put in Rs 10 crore or refuse. If both fund, the company survives and is worth Rs 120 crore. If only one funds, the money is not enough: the company is sold for Rs 30 crore and the funder loses its Rs 10 crore. If neither funds, it is sold for Rs 30 crore. What are the equilibria?Multi-stage VCSeries A to C VC
Try it first
Which outcomes are stable, in the sense that neither investor gains by changing its choice alone?
Show the worked solution
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.
Both funding pays each investor Rs 26 crore and both refusing pays Rs 9 crore, and each is stable because the lone funder ends at minus Rs 1 crore, so the bridge needs coordination rather than better economics. 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..
The relationshipq the chance you put on the other investor funding 26 your payoff if both fund, in Rs crore -1 your payoff if you fund alone 9 your payoff from refusing, whatever the other does What it says in wordsFunding beats refusing whenever you believe the other investor will fund with a chance above about 37%.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.
Where candidates lose it
The common loss is naming only both funding as the equilibrium because it pays the most. Equilibrium is about best replies, not about the best total, and both refusing is just as stable.
The second loss is stopping at the two equilibria without saying what to do about it. The interviewer wants to hear that an all-or-nothing commitment or a lead investor moving first removes the bad outcome, which is how bridges actually get done.
What the interviewer asks next
- If the lone funder got its Rs 10 crore back when the bridge fails, what are the equilibria now?
- How does a pay-to-play clause change each investor's payoffs?
- With three co-investors, each needed for the bridge, how does the trust threshold change?