Maximin Value Research Articles

Fair Allocation of Indivisible Goods: Improvement

We study the problem of fair allocation for indivisible goods. We use the maximin share paradigm introduced by Budish [Budish E (2011) The combinatorial assignment problem: Approximate competitive equilibrium from equal incomes. J. Political Econom. 119(6):1061–1103.] as a measure of fairness. Kurokawa et al. [Kurokawa D, Procaccia AD, Wang J (2018) Fair enough: Guaranteeing approximate maximin shares. J. ACM 65(2):8.] were the first to investigate this fundamental problem in the additive setting. They showed that in delicately constructed examples, not everyone can obtain a utility of at least her maximin value. They mitigated this impossibility result with a beautiful observation: no matter how the utility functions are made, we always can allocate the items to the agents to guarantee each agent’s utility is at least 2/3 of her maximin value. They left open whether this bound can be improved. Our main contribution answers this question in the affirmative. We improve their approximation result to a 3/4 factor guarantee.

Growth and long-run sustainability

AbstractFrom any state of economic and environmental assets, the maximin value defines the highest level of utility that can be sustained forever. Along any development path, the maximin value evolves over time according to investment decisions. If the current level of utility is lower than this value, there is room for growth of both the utility level and the maximin value. For any resource allocation mechanism (ram) and economic dynamics, growth is limited by the long-run level of the maximin value, which is an endogenous dynamic sustainability constraint. If utility reaches this limit, sustainability imposes growth to stop, and the adoption of maximin decisions instead of the currentram. We illustrate this pattern in two canonical models, the simple fishery and a two-sector economy with a nonrenewable resource. We discuss what our results imply for the assessment of sustainability in the short and the long run in non-optimal economies.

Deconstructing Generative Adversarial Networks

Generative Adversarial Networks (GANs) are a thriving unsupervised machine learning technique that has led to significant advances in various fields such as computer vision, natural language processing, among others. However, GANs are known to be difficult to train and usually suffer from mode collapse and the discriminator winning problem. To interpret the empirical observations of GANs and design better ones, we deconstruct the study of GANs into three components and make the following contributions. Formulation: we propose a perturbation view of the population target of GANs. Building on this interpretation, we show that GANs can be connected to the robust statistics framework, and propose a novel GAN architecture, termed as Cascade GANs, to provably recover meaningful low-dimensional generator approximations when the real distribution is high-dimensional and corrupted by outliers. Generalization: given a population target of GANs, we design a systematic principle, projection under admissible distance, to design GANs to meet the population requirement using only finite samples. We implement our principle in three cases to achieve polynomial and sometimes near-optimal sample complexities: (1) learning an arbitrary generator under an arbitrary pseudonorm; (2) learning a Gaussian location family under total variation distance, where we utilize our principle to provide a new proof for the near-optimality of the Tukey median viewed as GANs; (3) learning a low-dimensional Gaussian approximation of a high-dimensional arbitrary distribution under Wasserstein distance. We demonstrate a fundamental trade-off in the approximation error and statistical error in GANs, and demonstrate how to apply our principle in practice with only empirical samples to predict how many samples would be sufficient for GANs in order not to suffer from the discriminator winning problem. Optimization: we demonstrate alternating gradient descent is provably not locally asymptotically stable in optimizing the GAN formulation of PCA. We found that the minimax duality gap being non-zero might be one of the causes, and propose a new GAN architecture whose duality gap is zero, where the value of the game is equal to the previous minimax value (not the maximin value). We prove the new GAN architecture is globally asymptotically stable in solving PCA under alternating gradient descent.

Coalitions in Routing Games: A Worst-Case Perspective

We investigate a routing game that allows for the creation of coalitions, within the framework of cooperative game theory. Specifically, we describe the cost of each coalition as its maximin value. This represents the performance that the coalition can guarantee itself, under any (including worst) conditions. We then investigate fundamental solution concepts of the considered cooperative game, namely the core and a variant of the min-max fair nucleolus . We consider two types of routing games based on the agents’ Performance Objectives, namely bottleneck routing games and additive routing games . For bottleneck games, we establish that the core includes all system-optimal flow profiles and that the nucleolus is system-optimal or disadvantageous for the smallest agent in the system. Moreover, we describe an interesting set of scenarios for which the nucleolus is always system-optimal. For additive games, we focus on the fundamental load balancing game of routing over parallel links. We establish that, contrary to bottleneck games, not all system-optimal flow profiles lie in the core. However, we describe a specific system-optimal flow profile that does lie in the core and, under assumptions of symmetry, is equal to the nucleolus.

Unilateral Support Equilibrium, Berge Equilibrium, and Team Problems Solutions

We compare two notions of equilibrium for other-regarding agents, namely Berge and unilateral support equilibria. A Berge equilibrium is a strategy profile such that the teammates of each agent choose their strategies in order to maximize his utility. A unilateral support equilibrium is a strategy profile such that the teammates of each agent non-cooperatively choose their strategies to maximize his utility. By definition the level of cooperation in a unilateral support equilibrium is no higher than in a Berge equilibrium. Yet, relying on ideas from Team theory, we provide conditions under which a unilateral support equilibrium is also a Berge equilibrium. We also provide conditions under which a unilateral support equilibrium is a Berge–Vaisman equilibrium, i.e., a strategy profile which is a Berge equilibrium and such that the payoff of each player is no lower than his maximin value.

Sustainability of an economy relying on two reproducible assets

The highest utility level that can be sustained in an economy is given by the maximin value. To be able to use this value for sustainability accounting, the corresponding maximin problem must be solved. This paper studies the sustainability of an economy composed of two reproducible assets, each producing one of two consumption goods which are substitutes in utility. We characterize the maximin path of the economy, and associated maximin shadow values. We discuss how these shadow values could be used as accounting prices for development paths that differ from the maximin trajectory. The corresponding genuine savings indicator informs on the improvement or decline of the sustainable level of utility and the prospects of future generations.

Growth and Long-Run Sustainability

From any state of economic and environmental assets, the maximin value defines the highest level of utility that can be sustained forever. Along any development path, the maximin value evolves over time according to investment decisions. If the current level of utility is lower than this value, there is room for growth of both the utility level and the maximin value. For any resource allocation mechanism (ram) and economic dynamics, growth is limited by the long-run level of the maximin value, which is an endogenous dynamic sustainability constraint. If utility reaches this limit, sustainability imposes growth to stop, and the adoption of maximin decisions instead of the current ram. We illustrate this pattern in two canonical models, the simple fishery and a two-sector economy with a nonrenewable resource. We discuss what our results imply for the assessment of sustainability in the short- and the long-run in non-optimal economies.

On the Commitment Value and Commitment Optimal Strategies in Bimatrix Games

Given a bimatrix game, the associated leadership or commitment games are defined as the games at which one player, the leader, commits to a (possibly mixed) strategy and the other player, the follower, chooses his strategy after being informed of the irrevocable commitment of the leader (but not of its realization in case it is mixed). Based on a result by Von Stengel and Zamir [2010], the notions of commitment value and commitment optimal strategies for each player are discussed as a possible solution concept. It is shown that in nondegenerate bimatrix games (a) pure commitment optimal strategies together with the follower’s best response constitute Nash equilibria, and (b) strategies that participate in a completely mixed Nash equilibrium are strictly worse than commitment optimal strategies, provided they are not matrix game optimal. For various classes of bimatrix games that generalize zero-sum games, the relationship between the maximin value of the leader’s payoff matrix, the Nash equilibrium payoff and the commitment optimal value are discussed. For the Traveler’s Dilemma, the commitment optimal strategy and commitment value for the leader are evaluated and seem more acceptable as a solution than the unique Nash equilibrium. Finally, the relationship between commitment optimal strategies and Nash equilibria in [Formula: see text] bimatrix games is thoroughly examined and in addition, necessary and sufficient conditions for the follower to be worse off at the equilibrium of the leadership game than at any Nash equilibrium of the simultaneous move game are provided.

Resources in Parks and Police Management Applying Decision Utility to Solve Problems with Limited Resources

This is the first research article that attempts to relate public service to managing an organization and explains systems with realistic yet simplistic examples. This article is the first of its kind to relate public service to managing organizations that relate public service such as parks and police. It measures and implements maximin value functions. A maximin value function applies when the criteria are totally non-substitutable: a decrease in a critical criterion cannot be compensated for by an increase in another criterion. This article illustrates situations where a maximin value function is an appropriate model, develops a method to measure a decision maker's maximin value function, and demonstrates how a maximin value function can be used in applications such as park and police systems. The measurement technique is easy to understand and most decision makers can complete the process in a short period of time. For quantitative scheduling techniques found in journals, their wider use in applications has been declining due to a variety of obstacles. This article will first list a number of these obstacles and then suggest ways to overcome them. Parks and Police departments are government agencies that both have limited and competing resources. In these circumstances, it is an ideal situation to share the resources as much as possible. In this article, examples are shown of where the limited resources may occur in both agencies, and how the manager may overcome these problems by sharing the resources. Examples of affective and just sharing of resources are given for both parks and police departments. In Parks Management, affective trade-offs are shown among trim mowing, tractor mowing, garbage collection and ball-field dragging. In Police departments affective trade-offs are shown among foot patrol, car patrol, detective analysis and office work.

On maximin value and policy functions in an exhaustible resource model

AbstractThis paper studies maximin paths in the context of a standard exhaustible resource model. Under the assumption that the resource is important in production, it establishes the efficiency and uniqueness of non‐trivial maximin paths. It uses these results to study the nature of the maximin value and policy functions. The value function is shown to be differentiable with respect to the initial resource stock, and the derivative of the value function is related to the shadow prices associated with the maximin path starting from that resource stock. We show how maximin policy functions can be derived, using the maximin value function, and the fact that the maximin path always follows Hartwick's investment rule.

Maximin, minimax, and von Neumann–Morgenstern farsighted stable sets

In this note, we investigate the relationship between the classical concepts of maximin and minimax, which were originally defined in the context of zero-sum games in von Neumann and Morgenstern (1953), and the von Neumann–Morgenstern (vNM) farsighted stable set using the indirect domination defined in Chwe (1994). We show two main results for two-player games: an existence result and an almost-uniqueness result. Under a mild assumption, we show that any strategy profile that is Pareto efficient and strictly individually rational–that is, strictly above each player’s maximin value–is generically a singleton vNM farsighted stable set. Moreover, there does not exist a vNM farsighted stable set that includes a strategy profile that is strictly individually rational and yields a payoff greater than the minimax value for a player, but not Pareto efficient.

On the Fingerprinting Capacity Games for Arbitrary Alphabets and Their Asymptotics

The fingerprinting capacity has recently been derived as the value of a two-person zero-sum game. In this paper, we study the fingerprinting capacity games with k pirates in a new collusion model called the mixed digit model, which is inspired by the combined digit model of Škorić et al. For small k, the capacities along with optimal strategies for both players of the game are obtained explicitly. For large k, we extend our earlier asymptotic analysis for the binary alphabet with the marking assumption to q-ary alphabets with this general model and show that the capacity is asymptotic to A/(2k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ln q) where the constant A is specified as the maximin value of a functional game. Saddle-point solutions to the game are obtained using methods of variational calculus. For the special case of q-ary fingerprinting in the restricted digit model, we show that the interleaving attack is asymptotically optimal, a property that has motivated the design of optimized practical codes.

Value allocation under ambiguity

We consider a pure exchange economy with asymmetric information where individual behavior exhibits ambiguity aversion along the line of maximin expected utility decision making. For such economies, we introduce different notions of maximin value allocations. We also introduce a strong notion of incentive compatibility. We prove the existence and incentive compatibility of the maximin value allocation. We conclude that unlike the Bayesian value allocation approach in Krasa Yannelis (Econometrica 62(4):881–900, 1994), incentive compatibility is related to efficiency rather than to direct exchange of information.

An environmental-economic measure of sustainable development

A central issue in the study of sustainable development is the interplay of growth and sacrifice in a dynamic economy. This paper investigates the relationship among current consumption, sacrifice and sustainability improvement in a general context and in two canonical, stylized economies. We argue that the maximin value of utility measures what is sustainable and provides the limit to growth. Maximin value is interpreted as a dynamic environmental-economic carrying capacity and current utility as an environmental-economic footprint. The time derivative of maximin value is interpreted as net investment in sustainability improvement. It is called durable savings to distinguish it from genuine savings, usually computed with discounted-utilitarian prices.

Saddle functions and robust sets of equilibria

We provide a new sufficient condition for the robustness of sets of equilibria to incomplete information in the sense of Kajii and Morris (1997) [11], Morris and Ui (2005) [15]. The condition is formulated for games with a saddle function. A saddle function is a real-valued function on the set of action profiles such that there is a single player for whom minimizing the function implies choosing her best response, and for the other players maximizing the function implies choosing their best responses. In a game with a saddle function the set of correlated equilibria that induce an expectation of the saddle function greater or equal to its maximin value is robust to incomplete information.

An Environmental-Economic Measure of Sustainable Development

A central issue in the study of sustainable development is the interplay of growth and sacrifice in a dynamic economy. This paper investigates the relationship among current consumption, sacrifice, and sustainability improvement in a general context and in two canonical, stylized economies. We argue that the maximin value of utility measures what is sustainable and provides the limit to growth. Maximin value is interpreted as a dynamic environmental-economic carrying capacity and current utility as an environmental-economic footprint. The time derivative of maximin value is interpreted as net investment in sustainability improvement. It is called durable savings to distinguish it from genuine savings, usually computed with discounted-utilitarian prices.

An Algorithm for Computing Maximin of Mutual Informations

In this paper, we are concerned with the maximin of mutual informations, which usually occurs in the context of specifying the capacity of a compound channel or the maximum rate of broadcasting common messages to multiple receivers. Firstly, the analytical properties of the minimum mutual informations such as continuity, concavity and differentiability are analyzed. The necessary and sufficient conditions for the capacity-achieving input distributions are revealed by reformulating the original problem into an equivalent differentiable form. Also presented are bounds on the capacity. Secondly, an iterative method to compute the maximin value of two mutual informations is derived, which is almost the same as the Arimoto-Blahut algorithm except that a more complicated maximization algorithm is required at the second phase. Finally, the convergence of the proposed algorithm is proved.

Prudence, Commitments and Intertemporal Conflicts

AbstractTypical justifications of prudence are based on the fact that we are temporally extended agents who remain numerically identical over time. After showing that prudential considerations should instead be based on our identity at a particular time, I outline a normative context for prudential reasons, based on a present commitment to temporal neutrality. I then consider how contingency in the content of a present commitment to temporal neutrality provides a flexible context that can help to resolve current debates about whether agents are required to maximize value, maximin value, minimax loss or are required to conform to any of a host of other options on how to adjudicate various types of conflict between present and future values.

Learning to compete, coordinate, and cooperate in repeated games using reinforcement learning

We consider the problem of learning in repeated general-sum matrix games when a learning algorithm can observe the actions but not the payoffs of its associates. Due to the non-stationarity of the environment caused by learning associates in these games, most state-of-the-art algorithms perform poorly in some important repeated games due to an inability to make profitable compromises. To make these compromises, an agent must effectively balance competing objectives, including bounding losses, playing optimally with respect to current beliefs, and taking calculated, but profitable, risks. In this paper, we present, discuss, and analyze M-Qubed, a reinforcement learning algorithm designed to overcome these deficiencies by encoding and balancing best-response, cautious, and optimistic learning biases. We show that M-Qubed learns to make profitable compromises across a wide-range of repeated matrix games played with many kinds of learners. Specifically, we prove that M-Qubed's average payoffs meet or exceed its maximin value in the limit. Additionally, we show that, in two-player games, M-Qubed's average payoffs approach the value of the Nash bargaining solution in self play. Furthermore, it performs very well when associating with other learners, as evidenced by its robust behavior in round-robin and evolutionary tournaments of two-player games. These results demonstrate that an agent can learn to make good compromises, and hence receive high payoffs, in repeated games by effectively encoding and balancing best-response, cautious, and optimistic learning biases.

Highly Efficient Designs to Handle the Incorrect Specification of Linear Mixed Models

We apply a maximin criterion to examine the relative efficiency of several D q -optimal designs for a family of linear mixed models. Incorrect specifications of the order of the polynomial, size of the autocorrelation parameter, number of random parameters, and the correlation between random intercept and random slope are investigated. The results of our study allow us to draw the following conclusions: (1) the maximin D q -optimal design encountered appears to be highly efficient; (2) the variation of the minimum relative efficiencies of D q -optimal designs of the family of linear mixed models that were studied decreases as the order of the polynomial increases; (3) the effect of the autocorrelation parameter on the relative efficiencies of D q -optimal designs is the largest for first-degree polynomials; and (4) the relative efficiency of the equidistant design is lower than that of the maximin value and also lower than the reference value 0.85.