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Probabilistic Asymmetric Information and Lending Relationships. Philip Ostromogolsky Yale School of Management. Background. Often banks lend to small business customers over several periods.
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Probabilistic Asymmetric Information and Lending Relationships Philip Ostromogolsky Yale School of Management
Background • Often banks lend to small business customers over several periods. • Banks may offer a customer a first-time loan with the possibility that the customer may be able to get another, future loan from the bank if he does a good job repaying the first loan. • Over the period of the first loan the bank can monitor his borrower, learn about him, and use that information to extend or curtail future credit. • More information than is revealed by simply observing whether or not the customer repays his first-time loan.
How do banks compete over small business borrowers? • They bargain, just like stock brokers or shoppers at a public market. • This can be modeled as an English auction. • Banks are the bidders. • The potential small business borrower is the auctioneer.
I’m coming back to the White House What would Greenspan do? Ben Bill
A Simple Story About an Auction • The auctioneer says: • We will now conduct the auction, the highest bid ≥ 0 wins. • There is some probability p*[0,1] that the box contains a $100 bill. • I am not going to publicly disclose p*. • But, I will tell you that p* ~U[0,1] • I like Bill, so I am going to walk over to Bill and whisper in his ear the value of p*. • Ben is not going to be told anything about p*. • The auctioneer walks over to Bill and whispers the value of p* into his ear.
A Simple Story About an Auction • Bill is informed • Ben is uninformed • Bill makes the first bid • Bill’s bid = $0.00 • What should Ben do?
A Simple Story About an Auction • Ben Thinks: • Suppose probability that the box contains $100 = p* = 0.5, and of course Bill knows this. • Bill knows that the expected value of the box’s contents = 100p* = 50. • Bill will continue bidding up until Bill’s bid = 50. • If at some point Bill bids 50 and I then bid 51, I will win. • When the auctioneer announces that I have won my expected profit will = 100p* – my bid = 100*0.5 – 51 = 50 – 51 = -1. • So, when I am announced as the winner I will expect to have a profit of -1 < 0.
A Simple Story About an Auction • Ben Thinks: • If at some point I (Ben) bid 50, I will win. • When the auctioneer announces that I have won my expected profit will = 100p* – my bid = 100*0.5 – 50 = 50 – 50 = 0. • So, when I am announced as the winner I will expect to have a profit of 0. • If I ever bid some bid, Ben’s bid < 50, I of course will not win. • Thus, if p* = 50 and I don’t know that, I can never win, and I might actually lose!!!
A Simple Story About an Auction • Ben Thinks: • As of right now, the last bid, was Bill’s bid = 0. • I don’t actually know p*. • The lowest I could bid is $1. • If p* > 0.01 then the best I could hope for would be go get a profit of π = 0. • If p* < 0.01, then my profit would = π = 100p* – 1 < 100*0.01 – 1 < 0 • And, I would lose money!!! • Pr(p* < 0.01) = 0.01. • So, if I bid $1, the expected value of my profits = E[π] = 0.01(100p* - 1) = p* - 0.01 < 0 !!!
A Simple Story About an Auction • Thus, Ben drops out of the auction • and Bill obtains the contents of the box for a winning bid of $0. • Bill’s interim expected profits from this game are thus E[π|p*] = 100p* – 0 = 100p* • Bill’s ex ante expected profits from this game are Ep*[E[π|p*]] = 100E[p*]– 0 = 50.
A Simple Story About an Auction • Bill’s knowledge of p* does not just let him make a more accurate forecast of the expected value of the contents of the box. • this is an old idea. • It also gives him a credible deterrence device, through which he can force his opponent to exit the auction, and ensure himself maximum possible profits. • this is new idea.
The Experiment • 4 Simultaneous Auctions run by Boudhayan, Foong Soon, Michael, and me. • Each auction had 3 bidders. • Selection of informed bidder, randomization of p* and realization of box contents performed using random draws of poker chips. • Induce risk neutrality by giving each student an initial endowment of 10,000 point. • Incentivize strategic behavior by offering prizes for the 3 students having the most aggregate profit.
