To: pd-games@math.uio.no A Theory of Fraud I've recently been thinking of the incentives for producer fraud and consumer caution in game theoretic terms. My underlying intuition goes something like this. If all firms were honest, then there would be no incentive for consumer caution, so it would pay for some firms to become fraudulent. Conversely, if all firms were dishonest, then consumers would be very cautious, and a firm could get a lot of business by establishing a reputation, strong warranties, and so on. Thinking about this in terms of mixed strategy Nash equilibrium, it looks like (given plausible parameter values) you would get an equilibrium with most firms being honest (but some fraudulent), and most consumers being cautious (with some not so). The social optimum presumably would be all firms honest, no consumers cautious; the case for regulation on economic grounds would then have to argue that the improvement relative to the normal mixed strategy equilibrium justifies the costs of regulation. The worst case scenario for regulation would be if it were ineffective in reducing fraud, but tricked consumers into reducing their level of caution, just aggravating the problem it is meant to solve. To put this intuition a little more formally: Let us imagine a four player game. There are two firms and two consumers. The firms can play honest or fraudulent; the consumers can play cautious or uncautious. Unfortunately, it is hard to draw payoff boxes in ASCII, so I'll just describe the payoffs. 1. An honest consumer gets a fixed amount (say 9) if either firm is honest. If both firms are dishonest, he gets -1. (The underlying idea is that the payoffs are 10 and 0 respectively, with a cost of caution of 1.) 2. A dishonest consumer gets a payoff determined by the mix of honest and dishonest firms. If both firms are honest, he gets 10; if both are dishonest, he gets -30; if there is a mix, then he gets alpha*10+(1-alpha)*-30, where alpha is the fraction of honest firms. 3. If both firms are honest, then they both get 10. 4. If both firms are dishonest, then they get (expectationally) 12 for each uncautious consumer. 5. If one firm is honest and the other is dishonest, then the honest firm gets all of the business of the cautious consumers (10 per cautious consumer) and 50% of the business of the uncautious consumers (5 per uncautious consumer, taking expectations). The fraudulent firm gets 50% of the uncautious consumers, taking each of them for 24 (expected value 12 per uncautious consumer). Now if I remember everything correctly, then under normal parameter values you get the expected mixed strategy equilibrium that I describe. I don't think that any pure strategy exists unless you fiddle a lot with the parameters. To extend this to a more realistic setting, think of extending this to a game of many firms and consumers; the mixed strategy result becomes especially intuitive then. Any comments? Bryan Caplan Department of Economics Princeton University From math-pd-games-request@ulrik.uio.no Wed Jan 18 05:41:06 1995 Received: from pat.uio.no by ponyexpress.princeton.edu (5.65c/1.7/newPE) id AA25839; Wed, 18 Jan 1995 05:41:04 -0500 Received: from motgate.mot.com by pat.uio.no with SMTP (PP) id <13792-0@pat.uio.no>; Wed, 18 Jan 1995 11:39:20 +0100 Received: from mothost.mot.com by motgate.mot.com with SMTP (5.67b/IDA-1.4.4/MOT-3.1 for ) id AA27794; Wed, 18 Jan 1995 04:39:15 -0600 Received: from hpux4.miel.mot.com by mothost.mot.com with SMTP (5.67b/IDA-1.4.4/MOT-3.1 for ) id AA28697; Wed, 18 Jan 1995 04:39:09 -0600 Received: by hpux4.miel.mot.com id AA22859 (5.67b/IDA-1.5 for pd-games@math.uio.no); Wed, 18 Jan 1995 16:09:29 +0500 Date: Wed, 18 Jan 1995 16:09:29 +0500 From: Sanjeev N Khadilkar Message-Id: <199501181109.AA22859@hpux4.miel.mot.com> To: pd-games@math.uio.no Subject: A Theory of Fraud Status: RO Hi, Byran! In clause 3, does each firm get 10 per consumer or (expectationally) 5 per consumer? The reason this is interesting is as follows: Assume the second interpretation for clause 3; i.e. if both firms are honest, each firm gets (expectationally) 5 per consumer. Let beta be the fraction of cautious consumers. Consider the following cases: Case A: Both firms are honest. Then each firm gets 5 per consumer regardless of the value of beta. Case B: Both firms are dishonest. Then each firm gets (1 - beta) * 12 per consumer (expectationally). Case C: Firm F1 is honest and firm F2 is dishonest. Then F1 gets (beta * 10 + (1 - beta) * 5) per consumer (expectationally). F2 gets (1 - beta) * 12 per consumer (expectationally). Now add up the profits of both firms put together in each of these cases. In Case A, the total is 5 + 5 = 10 per customer. In Case B, the total is (1 - beta) * 24 = (24 - beta * 24). In Case C, the total is (beta * 10 + (1 - beta) * 17)= (17 - beta * 7). Compare Case A with Case C. Since beta ranges from 0 to 1, the total in Case C CAN NEVER BE LESS than the total in Case A. To put it another way, if the two firms agree to pool and split their takings, then total dishonesty puts them at the mercy at the consumer, whereas by playing a hard_guy-soft_guy game, they can never do worse than total honesty REGARDLESS OF THE STRATEGY FOLLOWED BY THE CONSUMERS (and stand to gain if there is at least one uncautious consumer). Of course, this holds only if there is cooperation of all firms playing the market and entry into the market is too difficult for new firms. Such a situation may hold in elections where there are only two big established candidates and new candidates have things stacked against them. Sanjeev N. Khadilkar snkhad@hpux4.miel.mot.com > From: Bryan Douglas Caplan > Date: Tue, 17 Jan 95 09:35:57 EST > Message-Id: <9501171435.AA21652@flagstaff.Princeton.EDU> > To: pd-games@math.uio.no > > A Theory of Fraud > > > Let us imagine a four player game. There are two firms and two > consumers. The firms can play honest or fraudulent; the consumers > can play cautious or uncautious. Unfortunately, it is hard to draw > payoff boxes in ASCII, so I'll just describe the payoffs. > 1. An honest consumer gets a fixed amount (say 9) if either > firm is honest. If both firms are dishonest, he gets -1. (The > underlying idea is that the payoffs are 10 and 0 respectively, > with a cost of caution of 1.) > 2. A dishonest consumer gets a payoff determined by the mix of > honest and dishonest firms. If both firms are honest, he gets 10; > if both are dishonest, he gets -30; if there is a mix, then he > gets alpha*10+(1-alpha)*-30, where alpha is the fraction of honest > firms. > 3. If both firms are honest, then they both get 10. > 4. If both firms are dishonest, then they get (expectationally) > 12 for each uncautious consumer. > 5. If one firm is honest and the other is dishonest, then the > honest firm gets all of the business of the cautious consumers > (10 per cautious consumer) and 50% of the business of the > uncautious consumers (5 per uncautious consumer, taking expectations). > The fraudulent firm gets 50% of the uncautious consumers, taking > each of them for 24 (expected value 12 per uncautious consumer). > > > Any comments? > > Bryan Caplan