Sjur Didrik Flåm

University of Bergen | UiB · Informatics Department

Can order markets lead participants towards price-taking equilibrium? Viewing market sessions as steps of iterative algorithms, this paper indicates positive prospects for convergence. Mathematical arguments turn on convolution, efficiency and generalized gradients. Economic arguments revolve around reservation costs, derived from indifference or t...

This paper considers common use of natural, renewable resources. It identifies good prospects for efficiency and welfare. To be precise, a core outcome -- hence cooperation -- can be secured over time by principal planning of total quotas, and in time by agents who share these in short-term markets. Information flows in two directions: to the princ...

By the first welfare theorem, competitive market equilibria belong to the core and hence are Pareto optimal . Letting money be a commodity, this paper turns these two inclusions around. More precisely, by generalizing the second welfare theorem we show that the said solutions may coincide as a common fixed point for one and the same system. Mathema...

Most microeconomic and game theoretic models of individual choice overlook adjustment costs. Rather often, the modeler’s concern is just with improvement of objectives. This optic doesn’t quite fit agents somewhat tied to status quo. If rational, any such agent reasons whether moving to another state be worth his while. For that, the realized gains...

Motivated by management problems in national fisheries, we examine management of renewable resources in local or regional commons. This paper suggests that property rights, or lack thereof, be replaced by well-defined user rights. It shows that the use of commons can be conditioned, paid for, or valued, via market mechanisms. To that end, direct de...

Price-taking behavior is the bedrock of much market theory. How might such behavior emerge? Addressing that old but still intriguing question, this paper uses a money commodity to denominate all rates of exchange and substitution. Out of equilibrium, some rates differ between agents, thereby driving trade. The simplest form of trade is bilateral; i...

This paper considers economies in which each agent valuates various goods by own generalized gradients. Taken together and appropriately scaled, the latter determine bid–ask spreads. When all such spreads are nil, market equilibrium prevails. Crucial for the arguments is a money commodity which denominates agents' rates of exchange or substitution....

Considered here are extremal convolutions concerned with allocative efficiency, risk sharing, or market equilibrium. Each additive term is upper semicontinuous, proper concave, maybe non-smooth, and possibly extended-valued. In a leading interpretation, each term, alongside its block of coordinates, is controlled by an independent economic agent. V...

Many mathematical models of strategic play or better choice overlook adjustment costs. Rather often, the modeler's concern is just with improvement of objectives. This optic doesn't quite fit agents somewhat attached to status quo. They reason whether moving to another state be worth their while. For that, the realized gains must outweigh the incon...

Suppose each member of some syndicate applies a monetary measure to price risk. Then, how might they reasonably share risk? What premiums could apply to insurance policies? More basically: can modestly informed, moderately skilled members eventually allocate risk efficiently and fairly? These questions are framed here below by convoluting the membe...

Considered here is repeated interaction among economic agents. These must share privately held user rights to diverse production factors. The disparate features of the resulting economy motivate a solution concept which blends Cournot/Nash equilibrium with that of Walras. A novelty comes by showing that integrated equilibrium may emerge via adaptiv...

This paper considers how an order market might evolve over a fairly short period – say, during a day. It uses elementary convex analysis to model agents’ choice of prudent orders, and it explores whether equilibrium is attainable.

Economic theory relates prices to quantities via ” market curves.” Typically, such curves are monotone, hence they admit functional representations. The latter invoke linear pricing of quantities so as to obtain market values. Specifically, if higher prices call forward greater supply, a convex function, bounded below by market values, represents t...

Many noncooperative settings require sharing of aggregate holdings—be these of natural resources, production tasks, or pollution permits. This paper considers instances where the shared items eventually become competitively priced. For that reason, the solution concept incorporates features of Nash and Walras equilibria. Focus is on how the concern...

If a car, already on the road, is replaced by another one, more expen- sive to collide with, a negative externality spills over to other drivers. This paper studies such externalities, relating them to insurance and incentives. It formalizes links from liability rules to choice of car. By assumption, insurance is cooperative but car acquisition is...

The distinguished econometrician Ragnar Frisch (1895–1973) also played an important role in optimization theory. In fact, he was a pioneer of interior-point methods. This note reconsiders his contribution, relating it to history and modern developments.

Motivated by computerized markets, this paper considers direct exchange between matched agents, just two at a time. Each party holds a ”commodity vector,” and each seeks, whenever possible, a better holding. Focus is on feasible, voluntary exchanges, driven only by (projected) differences in generalized gradients. The paper plays down the importanc...

Monetary risk measures are studied here in terms of acceptable outcomes, normalized non-negative prices, and resulting shortfalls. At center stage stand convex analysis, saddle functions and associated max-min formulae. The latter comply with common sense and established theory.

Considered here is direct exchange of production allowances or input factors. Motivated by practical modelling and compution, we sup- pose every owner or user of such items has a linear technology. The issue then is whether competitive market equilibrium can be reached merely via iterated bilateral barters. This paper provides positive and construc...

Outlined here are the research papers published in Volume 1 of the special issue Stochastic Financial Economics, each considering investor behavior in financial markets.

Exchange is modelled here as iterated bilateral barters, each fairly myopic and driven merely by gradient differences. Under weak conditions, repeated transactions carry the economy to market equilibrium, supported by clearing prices. A main feature is that agents, in the interim, are allowed non-admissible, possibly speculative, but sharply penali...

Projects, private or public, that share input factors or output requirements had better be construed as members of a portfolio. Present risk, the capital asset pricing model may facilitate valuation of each member. Chief results of that model are derived and generalized here as core solutions to a transferable-utility production game. Shadow prices...

When investors or reinsurers measure economic risk in monetary terms, they operate as though utility were transferable. A main purpose of this paper is to show that transferability largely facilitates attainment, analysis, computation and modelling of equilibrium in exchange economies. To wit, under reasonable and weak assumptions, it is shown that...

Outlined here are the research papers published in Volume 2 of the special issue Stochastic Financial Economics, each dealing with convex risk measures.

Motivated by repeated play of non-cooperative games, we study equation solving undertaken in parallel by several non-communicating agents, each dealing with his own block of variables. The process is akin to Newton's method in using derivative information. It does, however, proceed without matrix inversion and dispenses with the need to exchange in...

Considered here are equilibria, notably those that solve noncooperative games. Focus is on connections between evolutionary stability, concavity and monotonicity. It is shown that evolutionary stable points are local attractors under gradient dynamics. Such dynamics, while reflecting search for individual improvement, can incorporate myopia, imperf...

This note considers production (or market) games with transferable utility. Prime objects are explicitly computable core solutions, or somewhat "deficit" versions of such, fully defined by shadow prices. Main ar- guments revolve around standard Lagrangian duality. A chief concern is to relax, or avoid, the commonplace assumption that all preference...

Catch and processing capacity in Norwegian industrial fisheries may be extensively reduced without affecting gross revenue. A linear programming model for maximizing annual revenue in the Norwegian industrial fisheries was formulated. The main purpose of the modeling effort was to determine the minimal size of the fleet and the industry. The constr...

Considered here is on-line portfolio management aimed at maximizing the long-run growth of financial wealth. The portfolio is repeatedly rebalanced in response to observed returns on diverse assets. Suppose statistical information and related methods are not available—or deemed too difficult. On that assumption this paper explores how an adaptive p...

Exchange of contingent claims is construed here as a cooperative game with transferable utility. Solutions are sought in the core. The novelty is that agents, being uncertainty averse, may use distorted, subjective probabilities. Choquet integrals therefore replace expected utility. When convoluted payoff is concave at the aggregate endowment, ther...

We consider transferable-utility, cooperative games, featuring differently informed players. Parties can exchange endowments or undertake joint production, but not pool information. Coalitional contracts must therefore comply with members’ private information. Qualitatively different shadow prices then arise: some for material endowments, others fo...

It is common to tolerate that a system’s performance be unsustainable during an interim period. To live long however, its state must eventually satisfy various constraints. In this regard we design here differential inclusions that generate, in one generic format, two distinct phases of system dynamics. The first ensures feasibility in finite time;...

Advocated and illustrated here is stochastic approximation. That tractable tool produces quantitative estimates apt to illuminate pros and cons of various market regimes.

The paper considers approximations of convex programs involving expectation functionals. These approximations are based on conditional expectations and yield variational convergence for all relevant topologies on the decision space.

Criticism of expected utility theory emphasizes the asymmetry between gains and losses. Also stressed is the role of actual wealth. These aspects invite special scrutiny of risk aversion, whether in the small or in the large, at a reference point, called the status quo, where utility is non-smooth.

This paper explores a few cooperative aspects of investments in uncertain, real options. By hypothesis some production commitments, factors, or quotas are transferable. Cases in point include energy supply, emission of pollutants, and harvest of renewable resources. Of particular interest are technologies or projects that provide anti-correlated re...

Many economic models and optimization problems generate (endogenous) shadow prices—alias dual variables or Lagrange multipliers. Frequently the “slopes” of resulting price curves—that is, multiplier derivatives—are of great interest. These objects relate to the Jacobian of the optimality conditions. That particular matrix often has block structure....

This paper considers a fairly large class of noncooperative games in which strategies are jointly constrained. When what is called the Ky Fan or Nikaidô-Isoda function is convex-concave, selected Nash equilibria correspond to diagonal saddle points of that function. This feature is exploited to design computational algorithms for finding such equil...

Considered here is decentralized exchange of privately owned commodity bundles. Voluntary transactions take the form of repeated bilateral barters. Under broad and reasonable hypotheses the resulting process converges to competitive equilibrium. Price-taking behavior is not assumed. Prices emerge over time; they need neither be anticipated nor know...

Risk exchange is considered here as a cooperative game with transferable utility. The set-up fits markets for insurance, securities and contingent endowments. When convoluted payoff is concave at the aggregate endowment, there is a price-supported core solution. Under variance aversion the latter mirrors the two-fund separation in allocating to eac...

Choice of contingent claims could reflect risk aversion or pessimism. Accordingly, the underlying, but hidden preferences might fit expected utility of customary von Neumann-Morgenstern form - or more generally, comply with a Choquet integral. This paper considers constrained choice and rationalizes both sorts of attitudes. Two avenues are pursued:...

Financial options typically incorporate times of exercise. Alternatively, they embody set-up costs or indivisibilities. Such features lead to planning problems with integer decision variables. Provided the sample space be finite, it is shown here that integrality constraints can often be relaxed. In fact, simple mathematical programming, aimed at a...

This paper considers a fairly large class of noncooperative games in which strategies are jointly constrained. When what is called the Ky Fan or Nikaido-Isoda function is convex-concave, selected Nash equilibria correspond to diagonal saddle points of that function. This feature is exploited to design computational algorithms for finding such equil...

Nonsmooth analysis and exact penalty methods give much prominence to functions that exit from certain lower-level sets with positive upward slope. A novel, weak version of such slopes is formalized here and studied in terms of differential properties. Applied to inf-convolutions, those properties help to recover and extend recent results on nonconv...

Focus is here on coalitional games among economic agents plagued by aggregate pollutions of diverse sorts. Any contracting player presumably pollutes less than if he defects. In addition, cooperation among some parties benefits the outsiders. Then, granted convex preferences and technologies, the core is proven nonempty. Also, under natural assumpt...

The paper is concerned with a concave, infinite horizon, discrete time, stochastic optimization model. We characterize optimal solutions in terms of dual variables and prove that under appropriate conditions of strict concavity, all optimal trajectories will approach each other in distribution irrespective of starting point.

This paper applies a continuous-time version of the subgradient projection algorithm to find equlibria of non-cooperative games. Under monotonicity assumptions this algorithm is known to generate trajectories which Cesaro converge weakly to the solution set. Convergence in norm is established under a strict montonicity assumption. Strong monotonici...

We exhibit a continuous-time adjustment process in cooperative games (on characteristic form) which, under broad hypothesis, is shown to converge towards core solutions. The process lends itself to a natural interpretation: While the payoff is still outside the core, at least some dissatisfied player(s) succeeds in marginally improving his lot. Exa...

This paper deals with approximation schemes for infinite horizon, discrete time, stochastic optimization problems. We construct finite horizon approximates that yield upper and lower estimates and whose optimal solutions converge to long-term optimal trajectories. The results extend those of [3] from the deterministic case to the stochastic.

We consider an oligopolistic industry extracting a non-renewable resource sold in a competitive market. We show, first, that if all players, but one, have infinite private reserves and production capacities (or alternatively the expansion of the production capacities is exogeneous) then Nash open-loop and feedback equilibria coincide. Second, we sh...

This paper deals with project evaluation from a portfolio perspective. The chief motivation stems from pricing bundles of related projects, all affected by uncertainty, when markets are imperfect or absent. Novelties come by construing single projects as “players” of a transferable utility, stochastic, cooperative game. Stochastic programming then...

This paper studies measurement of welfare e¤ects, transient and permanent, of stabilizing or deregulating prices in Cobweb-like settings. As in Cobweb-models, producers must commit inputs in face of uncertainty. Here, however, we consider producers who are concerned with adaptations of inputs rather than price predictions. This shift of emphasis re...

This note considers production (or market) games with transferable utility. It brings out that, in many cases, explicit core solutions may be defined by shadow prices — and reached via quite natural dynamics.

We show that, in cooperative production games, when the production functions are not concave, the core may well be empty. However, as the number of players increases (subject to some regularity conditions), the relative deficit obtained by using concavified functions decreases to zero. Furthermore, differentiability of the functions will cause the...

The main objects here are finite-strategy games in which entropic terms are subtracted from the payoffs. After such subtraction each Nash equilibrium solves an explicit, unconstrained, nonlinear system of smooth equations. That system, while characteristic of perturbed best responses, is amenable in computation. It also facilitates analysis of fict...

This paper looks at social insurance of short term absence from work. The chief concern is with efficiency properties of full coverage. That arrangement is reviewed and criticized here in light of received theory. A main point is that positive loading of the premium implies less than full coverage. Concerns with optimal risk sharing also pull in th...

A feasibility problem typically amounts to locate at least one point which belongs to several closed subsets of a Euclidean space. Equivalently, one might seek a vector which minimizes several associated objectives. We develop a method to solve a class of such problems, allowing here non-convex sets or functions. The method uses proximal point iter...

This paper explores some cooperative aspects of investments in uncertain, real options. Key production factors are assumed transferable. They may reflect property or user rights. Emission of pollutants and harvest of renewable resources are cases in point. Of particular interest are alternative projects or technologies that provide inferior but ant...

Nash equilibrium leaves the impression that each player foresees perfectly and responds optimally. Must human-like, rational agents really acquire both these faculties? This paper argues that in some in-stances neither is ever needed. For the argument repeated play is mod-elled here as a constrained, decentralized, second-order process driven by no...

The main objects here are games in which players mainly compete but nonetheless collaborate on some subsidiary activities. Play assumes a two-stage nature in that first-stage moves presume coordination of some subsequent tasks. Specifically, we consider instances where second-stage coordination amounts to partial cost sharing, anticipated and susta...

We explore optimal search for individual improvement when agents start with different confidence in their own ability. The initial self-confidence may be determined by nature or socioeconomic factors. Presuming Bayesian learning, we show that final achievements depend positively on initial confidence. When parents' achievements affect children's se...

Main objects here are stochastic programs, possibly non-convex. We develop an algorithm that combines gradient projection with the heavy-ball method. What emerges is a constrained, stochastic, second-order process. Some friction feeds into and stabilizes myopic approximations. Convergence obtains under weak and natural conditions, an important one...

There is given a market for several perishable goods, supplied under technological randomness and price uncertainty. We study whether and how producers eventually may learn rational price expectations. The model is of cobweb type. Its dynamics fit standard forms of stochastic approximation with either variable or constant stepsizes. Relying upon qu...

Stochastic programming offers handy instruments to analyze exchange of goods and risks. Absent efficient markets for some of those items, such programming may imitate or synthesize market-like transfers among concerned parties. Specifically, using shadow prices (Lagrange multipliers) on aggregate endowments, one may identify side-payments that yiel...

Main objects here are normal-form games, featuring uncertainty and noncooperative players who entertain local visions, form local approximations, and hesitate in making large, swift adjustments. For the purpose of reaching Nash equilibrium, or learning such play, we advocate and illustrate an algorithm that combines stochastic gradient projection w...

So-called potential functions are important, prominent, and common to many diverse fields, including optimization, dynamic processes, and physics. Monderer and Shapley have added a class of noncooperative games to that list. In the present paper, their notion is extended and repeated play of such games is considered. A unified convergence analysis...

This note deals with Cournot type oligopolies in which the market clearing price occasionally may be non-unique. A Stackelberg leading producer is present. Given that setting we explore continuity properties of the followers' reaction and provide sufficient conditions for existence of equilibrium.

A standard approach to duality in stochastic optimization problems with constraints in L ∞ relies upon the Yosida-Hewitt theorem. We develop an alternative technique which employs only “elementary” means. The technique is based on an e-regularization of the original problem and on passing to the limit as e→0 with the help of a simple measure-theore...

Theoretical and experimental studies of noncooperative games increasingly recognize Nash equilibrium as a limiting outcome of players' repeated interaction. This note, while sharing that view, illustrates and advocates combined use of convex optimization and differential equations, the purpose being to render equilibrium both plausible and stable.

Exchange of risks is considered here as a transferable-utility cooperative game. When the concerned agents are risk averse, there is a core imputation given by means of shadow prices on state-dependent claims. Like in finance, a risk can hardly be evaluated merely by its inherent statistical properties (in isolation from other risks). Rather, evalu...

Motivated by non-cooperative games we study repeated interaction among noncommunicating agents, each dealing with his block of variables, each moving merely on the basis of his marginal payoff and its most recent change. Adjustment or play thus unfolds in parallel. Constraints are accommodated. The main issue is convergence to (Nash) equilibrium.

This paper deals with on-line computation—or step-wise learning—of Pareto optimal insurance contracts. Our approach tolerates that the loss distribution might be unknown, intractable, or not well specified. Thus we accommodate fairly inexperienced parties. Losses are here simulated or observed, one at a time, and they cause iterated revisions of th...

This note deals with Cournot type oligopolies in which the market clearing price occasionally may be non-unique. A Stackelberg leading producer is present. Given that setting we explore continuity properties of the followers' reaction and provide sufficient conditions for existence of equilibrium.

Emission of uniformly dispersed greenhouse gases in construed here as a cooperative production game, featuring side-payments, quata exchange, uncertainty, and multi-period planning. Stochastic programming offers good instruments to analyze such games. Absent efficient markets for emissions, such programming may help to imitate market-like, price-ba...

The main objects here are two-stage games in which players first compete and subsequently collaborate. We consider instances where second-stage collaboration is anticipated and sustained in terms of core solutions.

Accomodated here is a measure space of economic agents, each regarded as a profit maximizing producer, each endowed with his technology and resource bundle. Pooling of private endowments generates a cooperative game with side payments.

This paper deals with on-line computation—or step-wise learning—of Pareto optimal insurance contracts. Our approach tolerates that the loss distribution might be unknown, intractable, or not well specified. Thus we accommodate fairly inexperienced parties. Losses are here simulated or observed, one at a time, and they cause iterated revisions of th...

We consider repeated interaction among several producers of a homogeneous, divisible good, traded at a common market. Demand is uncertain, and its law is unknown.

We consider financial contracts that are tradable in any quantities at fixed prices. A bundle of such contracts constitutes an arbitrage if it offers non-negative payoff in any future state, but commands negative present cost. This article brings together fairly recent results on how to find an arbitrage provided some exists. Otherwise, a state-con...

Owners of stochastic assets can pool their endowments to smoothen and insure individual payoffs across outcomes and time. We explore, in such a setting, how contingent shadow prices on aggregate resources can be used for three purposes: first, to design mutual contracts for risk averse agents; second, to quantify the malfunctioning of such contract...

Prime objects of this note are (I) excess demandgenerated by priee-taking economic agents, and (II) an alternative version of tàtonnementWe relate laws of demand, axioms of revealed preferences, and other notions of generalized monotonicity to “evolutionary stable” prices. Focus is on localstability of competitive equilibrium. Specifically, we esta...

A standard approach to duality in stochastic optimization problems with constraints in L(infinite) relies upon the Yosida-Hewitt theorem. We develop an alternative technique which employs only "elementary" means. The technique is based on an e-regularization of the original problem and on passing to the limit as e --> 0 with the help of a simple me...

We consider a two-period, one-good financial market, featuring variance-averse investors. Under fairly weak assumptions, like those imposed in the capital asset pricing model, we demonstrate how equilibrium may be approached and computed. As main argument we use the two-dimensionality of pricing and the Poincare-Bedixon theory on planar flows.

How should the benefits of the commons, say a publicly owned fishing resource, be distributed? A first possibility is equal division among the population. A second option is to distribute them among the people who actually exploit the resource in proportion to their activity level: this is the ""land to the tiller"" view. A third approach is the nu...

Keywords The Basic Model Results Lagrange Multipliers for Phase Constraints Lagrange Multipliers for Nonanticipativity Constraints Synthesis See also References