On a general solution of finite order difference equation with constant coefficients

summary:In the present paper we give new formulas for a general solution of the linear difference equation of finite order with constant complex coefficients without necessity of solving the characteristic equatio

Towards a resolution of the privacy paradox

This paper provides an explanation of the so-called privacy paradox and describes a more general informational ’irrelevance’ result. We show that in a large class of imperfect information dynamic games between the buyer, the seller, and privacy platforms, the buyer chooses not to bear any direct cost of protecting her privacy even if leakage of her information affects the prices she faces and hence her surplus from trade. More generally, we show that the informed party’s choice of privacy (mode of communication) is driven solely by the direct cost of talk rather than by the information such talk conveys: choosing between different privacy options, the buyer always chooses a cheapest option irrespective of its and its alternatives’ informational characteristics

Complementary cooperation, minimal winning coalitions, and power indices

We introduce a new simple game, which is referred to as the complementary weighted multiple majority game (C-WMMG for short). C-WMMG models a basic cooperation rule, the complementary cooperation rule, and can be taken as a sister model of the famous weighted majority game (WMG for short). In this paper, we concentrate on the two dimensional C-WMMG. An interesting property of this case is that there are at most $n+1$ minimal winning coalitions (MWC for short), and they can be enumerated in time $O(n\log n)$, where $n$ is the number of players. This property guarantees that the two dimensional C-WMMG is more handleable than WMG. In particular, we prove that the main power indices, i.e. the Shapley-Shubik index, the Penrose-Banzhaf index, the Holler-Packel index, and the Deegan-Packel index, are all polynomially computable. To make a comparison with WMG, we know that it may have exponentially many MWCs, and none of the four power indices is polynomially computable (unless P=NP). Still for the two dimensional case, we show that local monotonicity holds for all of the four power indices. In WMG, this property is possessed by the Shapley-Shubik index and the Penrose-Banzhaf index, but not by the Holler-Packel index or the Deegan-Packel index. Since our model fits very well the cooperation and competition in team sports, we hope that it can be potentially applied in measuring the values of players in team sports, say help people give more objective ranking of NBA players and select MVPs, and consequently bring new insights into contest theory and the more general field of sports economics. It may also provide some interesting enlightenments into the design of non-additive voting mechanisms. Last but not least, the threshold version of C-WMMG is a generalization of WMG, and natural variants of it are closely related with the famous airport game and the stable marriage/roommates problem.Comment: 60 page

A case for pay-as-bid auctions

Pay-as-bid (or discriminatory) auctions are frequently used to sell homogenous goods such as treasury securities and commodities. We prove the uniqueness of their pure-strategy Bayesian Nash equilibrium and establish a tractable representation of equilibrium bids. Building on these results we analyze the optimal design of pay-as- bid auctions, as well as uniform-price auctions (the main alternative auction format). We show that supply transparency and full disclosure are optimal in pay-as-bid, though not necessarily in uniform-price; pay-as-bid is revenue dominant and might be welfare dominant; and we provide an explanation for the revenue equivalence observed in empirical work

The Random Priority mechanism is uniquely simple, efficient, and fair

Random Priority is a popular mechanism used to allocate a set of objects to a set of agents without the use of monetary transfers. Random Priority is appealing because it satisfies desirable efficiency, fairness, and incentive properties. Is it the only mechanism with these properties? We answer this long-standing question in the positive: Random Priority is the unique mechanism that is Pareto efficient, symmetric, and obviously strategy-proof

Efficient bilateral trade

Can two parties reach an ex-post Pareto efficient trade agreement? The importance of the question was elucidated by Coase (1960), and Myerson and Satterthwaite (1983) provided a commonly accepted negative answer that such agreement is impossible when the parties are privately informed. We show that this negative answer depends on the assumption of quasi-linear preferences: efficient trade is possible if risk- aversion or wealth effects are sufficiently large or if agents’ utility is not too responsive to private information. Under empirically-grounded specifications of risk aversion and elasticity of trade, two parties can trade efficiently despite substantial asymmetry of information

Non-bossiness and first-price auctions

We show that the first-price auction with no reserve price is the essentially unique mechanism that is non-bossy, individually rational, and efficient in equilibrium. The first-price auction with optimal reserve price is the essentially unique mechanism that is non-bossy, individually rational, and revenue maximizing

A theory of simplicity in games and mechanism design

We study extensive‐form games and mechanisms allowing agents that plan for only a subset of future decisions they may be called to make (the planning horizon). Agents may update their so‐called strategic plan as the game progresses and new decision points enter their planning horizon. We introduce a family of simplicity standards which require that the prescribed action leads to unambiguously better outcomes, no matter what happens outside the planning horizon. We employ these standards to explore the trade‐off between simplicity and other objectives, to characterize simple mechanisms in a wide range of economic environments, and to delineate the simplicity of common mechanisms such as posted prices and ascending auctions, with the former being simpler than the latter

Obviously strategyproof mechanisms in general environments

We consider the problem of characterizing all obviously strategy-proof (OSP) mechanisms for general preference environments. We show that any OSP mechanism is equivalent to a generalized millipede game in which agents are sequentially offered a menu of payoffs they may clinch (and thus leave the game), plus possibility the opportunity to pass (and remain in the game, hoping for better clinching options in the future). Our preference setting unifies many canonical mechanism design settings, such as single-unit auctions, public goods problems, and object allocation, and thus, many of the known OSP mechanisms are special cases of generalized millipede games. We also introduce other examples that fit our preference model that are new to the literature

Information choice: cost over content

When supplying information, agents choose between options that differ both in their contents and in their costs. We establish a “cost-over-content” theorem for a large class of dynamic trading environments where buyers choose from arbitrary sets of processes (experiments) that reveal information to the seller. When all experiments are equally costly, choosing any given experiment is a perfect equilibrium. However, when experiments differ in costs, there is a unique equilibrium: all buyers choose the cheapest experiment, regardless of the information it pro- vides. We explore implications for market performance, privacy, data sale, and defaults in market regulation