# How do I best prevent cheating using a credible threat in a two-person, non-cooperative, positive sum game?

I'm playing around with a situation in which there are two possible scenarios.

Scenario 1

This involves a transaction between two persons, Supplier and Consumer, who have the option of either behaving honestly or behaving corruptly. One thing to note is that a corrupt transaction is only possible when the Consumer is corrupt. If the Consumer is honest, the transaction will always be honest. The payoff matrix is shown below. (For convenience, I call the agents Supplier and Consumer. Don't worry about what these mean).

(In the images that follow, the Nash equilibrium is highlighted in the green box. I've highlighted the Supplier's best strategy and the Consumer's best strategy in RED and BLUE respectively).

Now comes the tricky part (tricky for me, that is). I'm trying to model an altered version of Scenario 1, called Scenario 2, described below.

Scenario 2

Here, there is a credible threat to the Supplier. The credible threat is that if both Supplier and Consumer act corruptly, and the Consumer gets caught (in my model, we can only detect corruption by the Consumer), then the Supplier will be punished, along with the Consumer. Basically, I want to discourage corruption even before it happens. If I only punished the Consumer, then Suppliers would feel free to latch onto the next corrupt Consumer that comes along. Since I am punishing both Supplier and Consumer, the Supplier might have to think twice before colluding with a corrupt Consumer.

I have a tentative payoff matrix below. I'd like to know if my reasoning in the matrix is correct.

I'm confused because there are two possible payoff matrices for Scenario 2. The only I've shown is the one in which they do get caught. What about the situation where they don't get caught? Do we need to take this into account? And most critical to me, is my reasoning (in italics above) about the double punishment (to Supplier and Consumer) in case of collusion being more advantageous for the system than simply punishing the Consumer a good strategy, assuming a system with a huge pool of Consumers and Suppliers?

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• Please clarify : you say that a corrupt transaction is only possible when the consumer is corrupt, but in the first payoff matrix we see non-zero payoffs when consumer is honest and supplier is corrupt. – Alecos Papadopoulos Dec 16 '17 at 15:50
• In Scenario 1, Honest-Honest is also a Nash equilibrium. – Michael Greinecker Dec 16 '17 at 18:36

Who makes that credible threat? What is the probability of being caught?

If the probability is certain, then there is only your second game. However, notice that you change in the 2nd game the best-reply structure, which means that now you have two equilibria, i.e., (H,H) and (H,C).

If the probability is less than 1, then you have two states of the world. One in which you are not caught (matrix 1) and the 2nd state of the world in where you are caught. Then that implies an analysis based on incomplete information.

So this depends on what is a credible threat. Because if the threat is credible, then the player should believe on it, and then a game as the one presented in scenario 2 is enough.

• Thanks. Yes, the threat would be credible with a certain probability, which is not equal to 1. If it were a definite credible threat, I'd have no problem with it, but since I'm dealing with a real world system, the threat has to be less than 1, even if it is high. I'm guessing that the way to make up for this fact is by making it very expensive for the ones getting caught, like by chucking them out of the system. How then would I model it? What is such a game called? It would involve probabilities and expectations, I'm guessing. – Joebevo Jun 18 '17 at 15:47
• You have a game with payoff uncertainty (static game of incomplete information). It depends on the probability that each player endows of being playing in each of the matrices. For example, for the consumer has a probability of playing scenario 1 with more than 1/2, then it compensates to play corrupt. If it is lower than 1/2, then is preferable to play honest. For the supplier you have to apply a similar reasoning. – user13650 Jun 18 '17 at 15:58
• Ok. But is there anything in game theory that takes into account the degree of the penalty? I can see what you're saying, but intuition seems to suggest that a stiffer penalty would ensure that I can police the players at lower cost to myself. – Joebevo Jun 18 '17 at 16:02
• If you want to learn a bit more about this kind of games, I would suggest you to read ch. 7 of the lecture notes of Battigalli academia.edu/1382053/Analysis_of_Strategic_Thinking – user13650 Jun 18 '17 at 16:02
• Depending on how much you aggravate the payoff on scenario 2 (C,C), less likely is that players will act in a corrupt way, by the same logic that I described before – user13650 Jun 18 '17 at 16:05

There aren't two payoff matrices. If there's a random choice between the two payoff matrices, then the overall payoff matrix will just be a linear combination of the two matrices you've presented, according to the probabilities. For instance, if the probability of getting caught is 10%, then each entry in the overall matrix is .9 times the corresponding entry in the first matrix, plus .1 times the entry in the second matrix. Since the two matrices are identical but for the bottom right, only that square will depend on the probability. If the probability of getting caught is $p$, then the payoff in that square will be $4(1-2p)$. More generally, if they have probability $p$ of getting a fine $f$ instead of the regular payoff, then the overall payoff will be $4(1-p)-pf=4-4p-pf=4-(f+4)p$. As you say, a higher probability of getting caught means less of a fine is required, and vice versa.

You have the payoffs for both players being equal, but suppose the Supplier and Consumer have different expected payoffs for the bottom right, SEP and CEP respectively. For the Supplier, any positive SEP will be an incentive to choose Corrupt. But for the Consumer, they need a CEP>2 to have an incentive to choose Corrupt. So we can remove the Nash Equilibrium by punishing just the Conusmer, and the required probability/size of the punishment for the Consumer will be lower than for the Supplier.

You say that even if you disincentivize the Consumer from Corrupt, the Supplier can go to another Consumer, but if the other Consumer has the same payoff matrix, then from a game theory point of view, as soon as we get a matrix for which the Consumer will choose Honest, the Supplier shouldn't be able to find any Consumer to be Corrupt.

Keep in mind that any game theory analysis will be based on only what's in the payoff matrix, so if there's some phenomenon that you think will affect the result, then for it to be included in the analysis, the payoff matrix needs to reflect it somehow. One way to analyze a party's ability to move on to another counterparty is opportunity costs. Suppose I have a painting, and one potential buyer might pay 100k, while another might pay 90k. You might think that the payoff for getting the first buyer to buy it is 100k, but by selling it to the first buyer, I'm giving up the opportunity to sell it to the second. So making sure the first deal goes through might be worth only 10k to me. Similarly, even if a Supplier gets 2 units of utility for selling something, if they're giving up the opportunity to sell it to someone else, then it might be appropriate to consider their payoff to be less than 2.

Besides subtracting a NE we don't like, we can also look at how we can add one we do like. Currently, the Supplier's choice matter only if the Consumer chooses Corrupt; otherwise, they are indifferent. Thus, the only way to induce them to choose Honest is by changing the payoffs when the Consumer chooses Corrupt. If the Supplier's payoff in the bottom left corner were less than 2, then they would have an incentive to choose Honest when the Consumer chooses Honest. This would make (Honest, Honest) a Nash equilibrium. So if the Supplier had a chance, no matter how small, of being punished every time they choose Corrupt, regardless of whether the Consumer chooses Corrupt, then the upper left would be a NE.

They would then be in a situation where are two NE. They would prefer the bottom right one, while you prefer that they be in the top left. Such a situation is heavily history-dependent: once they're in one NE, getting into another requires coordination. So one strategy would be to first put a lot of effort towards catching and punishing the Consumer, lowering the payoff in the bottom right and eliminating it as a NE. Once they're in the top left NE, you could reduce the enforcement. This would re-introduce the bottom right as a NE, but they would now be in the upper left NE and would have to coordinate getting to the other one. You can then put in some effort into punishing the Supplier whenever they choose Corrupt but the amount of effort to keep that a NE would be less than that need to keep (Honest, Honest) a NE would be less than that needed to keep (Corrupt, Corrupt) not a NE.

An even better solution would be adding a "Snitch" option; if Honest players had the option of snitching on Corrupt ones, and were rewarded for doing so, then that would make Corrupt much less attractive.