# Teorija igara i dizajn mehanizma

Vidov, Melita (2016) Teorija igara i dizajn mehanizma. Diploma thesis, Faculty of Science > Department of Mathematics.

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Language: Croatian

This thesis processed mechanism design topic which is based on game theory, divided due to the number of players. At the beginning we analyze mechanisms between a seller and a buyer. First we deal with giving prices to single indivisible good where the seller is a mechanism designer. With intuitive approach we estimate that the best for the seller to do is based on assessments of the buyer’s type, which is given by distribution function, to set the price and maximizes expected profit. We are interested in whether there is better mechanism. We introduce the concept of direct mechanism and incentive compatibility and individual rationality constraints that are necessary for easier analysis of the problem. Mitigating circumstance is that it turned out (for all chapters) that for any indirect mechanism where the buyer does not have to report true type, there is appropriate direct mechanism (revelation principle). Furthermore, we bring characterizations of mechanisms that were necessary in pursuit of optimizing mechanism. At the end it turned out that the beginning strategy was the best choice because the seller’s objective function is linear because there is no risk. After that, we have moved to nonlinear pricing i.e., pricing divisible good, for example sugar. The outcome of this complex problem isn’t so trivial because buyer’s utility is no longer linear because of further defined function ν which enables us to manipulate. Based on assumptions that distribution of random variable for reported type is regular, we have got effective way for setting mechanism that maximizes profit. At the end of the chapter we have processed an example that shows that the introduced function ν gave as a discount on the amount. The second chapter process Bayesian mechanism where in the game we look for Bayes-Nash equilibrium between players. Chapter starts with observing an auction of one indivisible good where the buyer gives his good on an auction for $N > 2$ agents. Then we define analogues of already mentioned distribution function, direct mechanism, revelation principle, only now in a case with multiple variables. Thus, we have to define additional probability function of the sale, expected transfer and utility that depends only on reported type of one agent. Most of the consequences of the characterization of properties is analog. With a similar analysis as in the previous chapter, we get that we will sell the good to the buyer with the largest positive value of the function “psi”, otherwise we won’t sell. At the end we have an example that shows how to distribute the likelihood of the sale of goods to customers. The last topic that we are dealing with are public goods. We have a community $N \geq 2$ of people who have to decide whether they will produce some indivisible good. Decision is made on the basis of cost of production and the amount of transfers that agents pay. We also define the property ex ante and ex post of budget balance, which essentially says that overall transfers must be greater than the cost of production. We conclude that these are two equivalent concepts, therefore we can choose whichever we want to use. First we maximize welfare of the community. Intuition leads us to first best mechanism which tells us that the good will be produce if the total sum of the types is greater than the cost of production. Unfortunately, it turns out that incentive compatible, individually rational mechanism can be built only in trivial cases. Using Kuch-Tucherov theorem we get second best mechanism for non-trivial cases. With a similar analysis we get optimizing mechanism for profit maximization. By observing the example we can see that profit maximization requires bigger types of agents from that one that maximizes the welfare.