Brainteaser help
I am having some trouble with this brainteaser from Crack's book.
Here is the question:
You are bidding for a firm whose unknown true value is uniformly distributed between 0 and 1. Although you do not know the true value S of the firm, you do know that as soon as people learn you made a bid the value will double to 2S. Your bid, however will be accepted only if it is at least as large as the original value of the firm. How do you bid to maximize expected payoff?
Here is my solution:
Payoff = 2S - B * p(0 = S = B)
p(0 = S = B) = integral(0 to B) of (1/(1-0)) = B
E[payoff] = (2Es - B ) * B Es = .5 so E[payoff] = (1-B)*B
Take the derivative and we get that 1 - 2B = 0 and that B = .5 at the max, so we should expect to bid .5 to optimize our payoff. +1/4 EV of payoff
However the book solution just says to bid anywhere from 0 to 1 and expect to break even. Is my analysis wrong? What is the right answer?
Just curious, who and what are u applying for? PM me.
Your payoff is wrong.
It should be intergral with respect to S, from zero to 1, (2S - B) * 1_(B>S)
Where 1_(B>S) = 1 if B>S, 0 otherwise
This simplifies to integral from zero to B of (2S -B) dS
Which is trivially zero, independent of B.
You can also draw some simple diagrams for a non-analytic proof.
This is not really a brainteaser, more of a mental exercise.
Sorry to revive this thread but I ran into an extremely similar question for a job interview.
Can somebody explain what this function actually is? Is it the probability B>S?
Agreed. Here's an alternative way:
Assuming that 2S = 1 from the wording of the problem, an iterative thought process:
Let's start with the maximum bid B = 1.
S = 0.5 (since 2S cannot be greater than 1), and Payoff (P) = 0. To maximize payoff, we must minimize B, so B has to be less than or equal to 2S. In particular, B=S, so that we maximize P. Thus we can eliminate all B in [0.5, 1], since for any B in [0.5, 1], there is an equal probability that an S exists such that P = 0.
So B 0.5. P >= 0 only if S = 0.25, so by the previous reasoning, we can eliminate all B in [0.25, 5). This leaves us with B 0.25.
Repeat the same pattern of reasoning, and B tends to zero. Thus E(P) = 0 regardless of the choice of B, and the equilibrium bid is B = 0. Note that B in (0,1] also gives E(P) = 0, so these are equally valid choices. B = 0 is a loss-minimizing choice. This reasoning only works because S is a uniform distribution.
A quick and dirty way to think about it is that if the only way to win the auction is B>=S, the only way to make a profit is B=2S, and S doubles as soon as you make a bid, B = 0 is the only reasonable answer, because your expected profits will be zero, i.e. - you break even.
hmmm, what if we change the payoff to 1.5*S or to 3S, do we still break even?
Bid close to or equal to 1.
what if the firm's value is 1, and you bid 1.1? Wouldn't that result in a payoff of 2*1 - 1.1 = 0.9? I'm not sure I understand the question correctly.
It's the indicator function. It takes the value 1 when B>S and is 0 otherwise.
Oh, that makes sense. Thanks!
Mollitia voluptatibus saepe officia asperiores id. Aliquam accusantium labore eligendi. Blanditiis fugiat aut blanditiis cumque odio atque.
Eos quos eveniet corrupti. Voluptatem adipisci ipsam ut.
Magnam quasi deleniti sequi quo at perferendis reprehenderit. Eveniet adipisci minus alias sequi aut. Nobis rerum quidem et quia. Sit atque facilis quia voluptatem et deleniti eum. Minus deleniti fuga quaerat illo.
Optio cum officiis natus animi. Repellat et consequatur facilis et aperiam repellendus.
See All Comments - 100% Free
WSO depends on everyone being able to pitch in when they know something. Unlock with your email and get bonus: 6 financial modeling lessons free ($199 value)
or Unlock with your social account...