Non-standard concave utility functions

The equivalence of markets and games concerns the relationship between two sorts of structures that appear fundamentally different -- markets and games. Shapley and Shubik (1969) demonstrates that: (1) games derived from markets with concave utility functions generate totally balanced games where the players in the game are the participants in the economy and (2) every totally balanced game generates a market with concave utility functions. A particular form of such a market is one where the commodities are the participants themselves, a labor market for example. But markets are very special structures, more so when it is required that utility functions be concave. Participants may also get utility from belonging to groups, such as marriages, or clubs, or productive coalitions. It may be that participants in an economy even derive utility (or disutility) from engaging in processes that lead to the eventual exchange of commodities. The question is when are such economic structures equivalent to markets with concave utility functions. This paper summarizes research showing that a broad class of large economies generate balanced market games. The economies include, for example, economies with clubs where individuals may have memberships in multiple clubs, with indivisibile commodities, with nonconvexities and with non-monotonicities. The main assumption are: (1) that an option open to any group of players is to break into smaller groups and realize the sum of the worths of these groups, that is, essential superadditivity is satisfied and :(2) relatively small groups of participants can realize almost all gains to coalition formation. The equivalence of games with many players and markets with many participants indicates that relationships obtained for markets with concave utility functions and many participants will also hold for diverse social and economic situations with many players. These relationships include: (a) equivalence of the core and the set of competitive outcomes; (b) the Shapley value is contained in the core or approximate cores; (c) the equal treatment property holds -- that is, both market equilibrium and the core treat similar players similarly. These results can be applied to diverse economic models to obtain the equivalence of cooperative outcomes and competitive, price taking outcomes in economies with many participants and indicate that such results hold in yet more generality.

Robust optimization searches for recommendations that are relatively immune to anticipated uncertainty in the problem parameters. Stochasticities are addressed via a set of discrete scenarios. This paper presents applications in which the traditional stochastic linear program fails to identify a robust solution—despite the presence of a cheap robust point. Limitations of piecewise linearization are discussed. We argue that a concave utility function should be incorporated in a model whenever the decision maker is risk averse. Examples are taken from telecommunications and financial planning.

This paper extends Merton’s continuous time (instantaneous) mean-variance analysis and the mutual fund separation theory. Given the existence of a Markovian state price density process, the optimal portfolios from concave utility maximization are instantaneously mean-variance efficient independent of the concave utility function’s form. The Capital Asset Pricing Model holds with the market portfolio induced by the growth optimal portfolio. The Markowitz-Tobin mutual fund separation is extended to include the lognormal assumption for asset prices as a special case. Closed form solutions to the expected utility maximization of terminal portfolio value are derived. We present an example in which the state price processes are specified as a multivariate geometric Brownian motion and the asset prices follow a multivariate diffusion process with relatively general parameters. JEL classification: C61; D92; G11; G12

We develop efficient algorithms to construct approximately utility maximizing mechanisms for a risk averse seller in the presence of potentially risk-averse buyers in Bayesian single parameter and multi-parameter settings. We model risk aversion by concave utility function. Bayesian mechanism design has usually focused on revenue maximization in a risk-neutral environment, and while some work has regarded buyers' risk aversion, very little of past work addresses the seller's risk aversion. We first consider the problem of designing a DSIC mechanism for a risk-averse seller in the case of multi-unit auctions. We give a poly-time computable pricing mechanism that is a (1−1/e−e)-approximation to an optimal DSIC mechanism, for any e>0. Our result is based on a novel application of correlation gap bound, that involves splitting and merging of random variables to redistribute probability mass across buyers. This allows us to reduce our problem to that of checking feasibility of a small number of distinct configurations, each of which corresponds to a covering LP. DSIC mechanisms are robust against buyers' risk aversion, but may yield arbitrarily worse utility than the optimal BIC mechanisms, when buyers' utility functions are assumed to be known. For a risk averse seller, we design a truthful-in-expectation mechanism whose utility is a small constant factor approximation to the utilty of the optimal BIC mechanism under two mild assumptions: (a) ex post individual rationality and (b) no positive transfers. Our mechanism simulates several rounds of sequential offers, that are computed using stochastic techniques developed for our DSIC mechanism. We believe that our techniques will be useful for other stochastic optimization problems with concave objective functions.

The optimal design of a water distribution network is a simulation-optimization task that should consider conflicts between different groups of stakeholders directly or indirectly. Investors and consumers are two groups of stakeholders with conflicting goals. Young’s bargaining method is a decision tool based on game theory that can help decision-makers to select one of the design alternatives by considering utilities of stakeholders. In this paper, the optimal design of two benchmark network problems (Two-loop and Hanoi networks) is considered with minimization of design cost and maximization of system efficiency, with respect to increasing hydraulic pressure. In this regard, decision alternatives are first determined by using a multi-objective, fast, messy genetic algorithm (MOFMGA). Young’s bargaining method is then applied with different combinations of utility functions of stakeholders. Results show that the use of the same utility functions for both stakeholders improves 63.23% and 24.47% of investor goals and 79.08% and 45.69% of consumer goals compared to the worst possible alternatives in the Two-loop and Hanoi networks, respectively. Moreover, both investor and consumer goals improve 6.19% and 7.14% in the Two-loop and 22.73% and 6.07% in the Hanoi network using a more concave utility function whose emphasis is on stakeholder utility, respectively.

The service identifier and connection identifier mapping theory was analyzed in a new two-layer universal network. Optimization models are proposed for the four types of mapping from service identifier to connection identifier in the pervasive service layer based on the utility maximization for services. For the nonlinear optimization problem, elastic services with concave utility function are firstly considered and the optimal flow rate for these services can be obtained. For the inelastic services with sigmoidal utility function, the optimal flow rate allocation can also be obtained if the capacities of some connections are provided with certain values. Furthermore, the model for inelastic services with QoS guarantee and the model for services with priority are also presented. Some numerical examples are given to verify the results obtained.

The Integrability Problem of Asset Prices

In the original position meta-bargain among political agents, each can achieve higher expected utility by binding himself to a unanimity rule, rather than risk the chance of being an excluded party in later period-by-period bargains. This argument is illustrated for a three-agent, constant-sum game where there are three reasonable voting rule options: 1) dictatorship rule; 2) majority rule; or 3) unanimity rule. Given concave utility functions, by Jensen's Inequality, each player would prefer the coalition inclusion guaranteed by the unanimity rule as opposed to the possible exclusion inherent in the other two options. However, once transactions costs are considered, a unanimity rule will create situations where all voters might agree in principle to a policy proposition, yet they will fail to reach a unanimous consensus.

Community preferences for deriving aggregate consumption de- mand has been used in international trade theory for many years. Authors like Samuelson [12], Gorman [5], Muellbauer [8], Milne [7], and others, have investigated the problem: under what conditions can individual preferences be aggregated so that there exists a community preference map from which an aggregate de- mand function can be derived. Eisenberg [3] has shown that if the individuals' preferences are homothetic and the distribution of income is fixed, then the aggregate demand function can be derived from a homogeneous and concave utility function. In relationship to E i s e n b e rg's results, Chipman [1] published an article demonstrating that also in terms of the integrability theory, using an approach pioneered by Samu- elson and perfected by Hurwicz and Uzawa, the aggregation of individual demand functions is possible so that the community de- mand function acts as if it were derivable from a single preference ordering. As a basic assumption, Chipman also requires that the consumers' preferences are homothetic. This paper is concerned with the aggregation of individual de- mand functions in terms of the revealed preference analysis. Based on hypotheses according to Uzawa [11], we want to show that if consumers' demand functions are linearly homogeneous in income

In many applications including interactive services and big data analytics, a timely result with a good match is often more valuable than a perfect yet delayed result. This fact can be utilized to improve the total utility gain of a cloud computing platform by allowing partial execution of jobs. A fundamental challenge, however, is that in many real environments, scheduling decisions have to be made online without knowledge about future jobs, which makes it difficult to choose between more valuable jobs with large deadlines and less valuable jobs that are more emergent. Moreover, jobs are often heterogeneous in their utilities, deadlines, and demands for different types of resources. In this paper, we study the problem of online scheduling for deadline-sensitive jobs with concave utility functions that can deliver partial results. We develop efficient online multi-resource allocation algorithms that achieve low competitive ratios for both continuous and discrete job models.

We analyze individual investment behavior among 822 young men and women that are members of 111 formal business groups in northern Ethiopia. We collected baseline data and investment data one year later combined with incentivized field experiments to obtain dis-aggregated risk preference data. We find that businesswomen on average invest significantly less at individual level than businessmen but Cohen’s d values for the gender difference are moderate in size. Women are found to have higher Constant Relative Risk Aversion (CRRA-r) coefficients (more concave utility function), to be more loss averse, but also to be more optimistic in their expectations (lower Prelec β) than men. Women were also poorer in non-land assets, came from more land-poor parents, and had lower incomes. The gender differences in risk attitudes and baseline endowments could explain some of but not all of the gender differences in investment behavior.

The Objective Money affects every aspect of life and yet its impact on a macro or micro level is not clearly understood. From an individual's point of view an efficient cash management policy could free resources for consumption which otherwise may have been wasted on either holding or transaction costs. But few models analyse a risk-averter's cash management decision and its impact on the money stock. Part of this deficiency can be attributed to the difficulties which arise from the non-linearities inherent in concave utility functions. The sheer complexity of modelling the dynamic evolution of variables which influence the cash management decision, and, the interaction between them has been another factor. The history of research into the demand for money is vast and has been an important feature in the evolution of macroeconomic theory. Numerous modelling approaches have been utilised to study the many properties of money, varying from general equilibrium analysis to micro-based models in which the agent behaves like a private optimiser. Important contributions have been made by eminent economists on how the money stock behaves. Fisher's quantity theory identity MV = PY; M is the nominal stock of money, V is the velocity of circulation, P is the price level, and Y is the volume of transactions or real income, which was developed in 1911 still features prominently in economic analysis. The velocity of circulation is assumed to be determined by an exogenous payments mechanism and therefore constant. Hence any change to the money stock yields neutral effects over the long run. Pigou (1917) changes this to include the consumer allocation problem, interest rates and wealth, which subsequently comes to form the basis of the Cambridge equation. These models set the tone for the literature which later followed from the various Classical schools arguing in favour of a passive monetary policy. Keynes in his General Theory of Employment, Interest and Money (1936) radically challenges this view by arguing that velocity was not constant, but varied with the price level and income, which, therefore, required an interventionist monetary authority. He divides the money stock into three components proposing that agents hold money for three very different reasons. The first he concludes is the transactions motive where agents hold money to satisfy planned expenditure. The second is the precautionary motive where money is held as a buffer stock to absorb any unanticipated expenditure shocks. The third is the speculative motive where agents hold money because it is an asset. At the time Keynes wrote his general theory real appreciations in the value of the nominal money stock were not uncommon. Therefore the role of money as a speculative asset was more important then, than it is now. Baumol (1952) and Tobin (1956) formalise the transactions motive by placing it within a dual asset optimisation framework. Agents in these models optimally determine their money stock by minimising the associated opportunity costs. Miller and Orr (1966) develop this further by introducing uncertainty through a discrete steady state random walk. By limiting the type of agent considered to be risk neutral, they effectively model the problem as a dual asset management exercise in which the agent optimises his utility of his wealth, similar to Tobin (1958). Constantinides and Richard (1978) model the cash management decision as a net present value problem. Increasing the time horizon reduces the frequency of transactions in which agents switch from cash to the interest earning asset or vice versa but increases their magnitude. Smith (1989) expands on this by allowing for interest rate uncertainty. A critical review of the current literature on the transactions money demand for money is presented in Chapter 3. The original objective of this thesis was to expand on Smith (1989) by developing a model that studied a risk-averter's cash management decision which included genuine aspects of risk and a discretely varying stochastic interest rate. The motive behind this was to study the impact of increased risk sensitivity on an agent's money demand function and also capture the discrete jumps which interest rates exhibit in the real world.

Optimising investments from elephant tourist revenues in the Maputo Elephant Reserve, Mozambique

Hatfield, Kominers, Nichifor, Ostrovsky, and Westkamp showed the existence of stable outcomes and competitive equilibria in a model of trading networks under the assumption that all agents’ preferences satisfy a condition called the full substitutes condition. In this paper, we extend their model by using discrete concave utility functions called twisted \(\hbox {M}^{\natural }\)-concave functions. We show that a valuation function of an agent is twisted \(\hbox {M}^{\natural }\)-concave if and only if the agent’s preference satisfies the generalized variant of the full substitutes condition. We also show that under the generalized full substitutes condition, there exist stable outcomes and competitive equilibria in the extended model and the set of competitive equilibrium price vectors forms a lattice. In addition, we discuss the connection among competitive equilibria, stability, and efficiency. Finally, we investigate the relationship among stability, strong group stability, and chain stability and verify these three stability concepts are equivalent as long as valuation functions of all agents are twisted \(\hbox {M}^{\natural }\)-concave.

An effective linear approximation method for separable programming problems