Accumulation Games Research Articles

Asymmetric Information in a Capital Accumulation Differential Game with Spillover and Learning Effects

This paper considers a capital accumulation game where the installation costs of investments are lowered by the firm’s own capital stock because of learning and by the competitor’s capital stock because of spillover effects. To properly understand the impact of the two capital stocks, we consider six information structures which differ in whether a firm takes into account that their competitor’s strategy depends only on time or on one or both capital stocks. We find that if firms are aware that their own capital stock makes investments of the competitor more efficient, a firm would invest less. Due to this effect, information only pays off if it is taken into account by both competitors, because otherwise the less informed and therefore less cautious player invests more and has a higher capital stock and revenues in the long run.

Stabilization of capital accumulation games

Stabilization of capital accumulation games

Channel coordination under Nash bargaining fairness concerns in differential games of goodwill accumulation

We study a supply chain coordination problem with an upstream manufacturer and a downstream retailer that have Nash bargaining fairness concerns. The evolution of consumer goodwill is assumed to be influenced by the final product's quality improvement level (from the manufacturer) and advertising effort level (from the retailer). We determine the optimal strategies by differential game models for centralized, fairness-neutral decentralized, and fairness-concerned decentralized channels under different power structures. A revenue- and cost-sharing contract is also developed to coordinate the fairness-concerned decentralized channels. We show that decision-makers will accept the proposed contract only if the revenue-sharing rate satisfies certain conditions. Comparison discussions, sensitivity analyses of some key parameters, and numerical studies are conducted to provide further insights. We observe that a dominant channel member's sensitivity to fairness is relatively more significant in the decision-making process and channel efficiency. Specifically, each member has a greater incentive to adjust the investment and pricing strategies in the channel that it dominates. Additionally, we find that when the supply chain members are fairness-neutral, no power structure exists that can simultaneously be the most beneficial to the channel with the greatest profit, the lowest selling price, and the most improved quality. However, when the supply chain members are fairness-concerned, conditions exist under which either the retailer- or manufacturer-dominated channel can exhibit optimum performance in all three aspects.

Capital accumulation game with quasi-geometric discounting and consumption externalities

This study introduces quasi-geometric discounting into a simple growth model of common capital accumulation that takes consumption externalities into account. We examine how present bias affects economic growth and welfare, and we consider two equilibrium concepts: the non-cooperative Nash equilibrium (NNE) and the cooperative equilibrium (CE). We show that the growth rate in the NNE can be higher than that in the CE if individuals strongly admire the consumption of others regardless of the magnitude of present bias. Contrary to the results in the time-consistent case, we show that, when present bias is incorporated, the welfare level in the NNE can be higher than that in the CE in the initial period. However, in later periods, this relationship can be reversed depending on the difference in the speed of capital accumulation.

Accumulation games on graphs

Accumulation games on discrete locations were introduced by Ruckle and Kikuta. The Hider secretly distributes his total wealth h ≥ 1 over locations 1,2,…,n. The Searcher confiscates the material from any r of these locations. The Hider wins if the wealth remaining at the n − r unsearched locations sums to at least 1; otherwise the Searcher wins. Their game models problems in which the Hider needs to have, after confiscation (or loss by natural causes), a sufficient amount of material (food, wealth, arms) to carry out some objective (survive the winter, buy a house, start an insurrection). The conjecture of Kikuta and Ruckle shows that there is always an optimal Hider strategy which places equal amounts of material on certain locations (and nothing on the rest) is still open and known to be hard. This article takes the hiding locations to be the nodes of a graph and restricts the node sets which the Searcher can remove to be drawn from a given family: the edges, the connected r‐sets, or some other given sets of nodes. This models the case where the pilferer, or storm, is known to act only on a set of close locations. Unlike the original game, our game requires mixed strategies. We give a complete solution for certain classes of graphs. © 2014 Wiley Periodicals, Inc. NETWORKS, Vol. 64(1), 40–47 2014

R&D competition versus R&D cooperation in oligopolistic markets with evolving structure

This paper considers investment behavior of duopolistic firms subject to technological progress. It is assumed that initially both firms offer a homogeneous product, but after a stochastic waiting time they are able to implement a product innovation. Production capacities of both firms are product specific. It is shown that firms anticipate a future product innovation by under-investing (if the new product is a substitute to the established product) and higher profits, and over-investing (in case of complements) and lower profits, compared to the corresponding standard capital accumulation game. This anticipation effect is stronger in the case of R&D cooperation. Furthermore, since due to R&D cooperation firms introduce the new product at the same time, this leads to intensified competition and lower firm profits right after the new product has been introduced. In addition, we show that under R&D competition the firm that innovates first, overshoots in new-product capacity buildup in order to exploit its temporary monopoly position. Taking into account all these effects, the result is that, if the new product is neither a close substitute nor a strong complement of the established product, positive synergy effects in R&D cooperation are necessary to make it more profitable for firms than R&D competition.

A Mean Field Capital Accumulation Game with HARA Utility

This paper introduces a mean field modeling framework for consumption-accumulation optimization. The production dynamics are generalized from stochastic growth theory by addressing the collective impact of a large population of similar agents on efficiency. This gives rise to a stochastic dynamic game with mean field coupling in the dynamics, where we adopt a hyperbolic absolute risk aversion (HARA) utility functional for the agents. A set of decentralized strategies is obtained by using the Nash certainty equivalence approach. To examine the long-term behavior we introduce a notion called the relaxed stationary mean field solution. The simple strategy computed from this solution is used to investigate the out-of-equilibrium behavior of the mean field system. Interesting nonlinear phenomena can emerge, including stable equilibria, limit cycles and chaos, which are related to the agent’s sensitivity to the mean field.

R&D Competition Versus R&D Cooperation in Oligopolistic Markets with Evolving Structure

This paper considers investment behavior of duopolistic firms subject to technological progress. It is assumed that initially both firms offer a homogeneous product, but after a stochastic waiting time they are able to realize a product innovation. Production capacities of both firms are product specific. It is shown that firms anticipate a future product innovation by under-investing (if the new product is a substitute to the established product) and higher profits, and over-investing (in case of complements) and lower profits, compared to the corresponding standard capital accumulation game. This anticipation effect is stronger in the case of R&D cooperation. Furthermore, since due to R&D cooperation firms introduce the new product at the same time, this leads to intensified competition and lower firm profits right after the new product has been introduced. In addition, we show that under R&D competition the firm that innovates first, overshoots in new-product capacity buildup in order to exploit its temporary monopoly position. Taking into account all these effects, the result is that, if the new product is neither a close substitute nor a strong complement of the established product, positive synergy effects in R&D cooperation are necessary to make it more profitable for firms than R&D competition.

Irreversible investment in oligopoly

We take a general perspective on capital accumulation games with open loop strategies, as they have been formalized by Back and Paulsen (Rev. Financ. Stud. 22, 4531–4552, 2009). With such strategies, the optimization problems of the individual players are of the monotone follower type. Consequently, one can adapt available methods, in particular the approach of Bank (SIAM J. Control Optim. 44, 1529–1541, 2005). We obtain consistency in equilibrium by proving that with common assumptions from the oligopoly literature on instantaneous revenue, equilibrium determination is equivalent to solving a single monotone follower problem. In the unique open loop equilibrium, only the currently smallest firms invest. This result is valid for arbitrary initial capital levels and general stochastic shock processes, which may be non-Markovian and include jumps. We explicitly solve an example, the specification of Grenadier (Rev. Financ. Stud. 15, 691–721, 2002) with a Levy process.

On Ruckle's Conjecture on Accumulation Games

In an accumulation game, the Hider secretly distributes his given total wealth h among n locations, while the Searcher picks r locations and confiscates the material placed there. The Hider wins if what is left at the remaining $n-r$ locations is at least 1; otherwise the Searcher wins. Ruckle's conjecture says that an optimal Hider strategy is to put an equal amount $h/k$ at k randomly chosen locations for some k. We extend the work of Kikuta and Ruckle by proving the conjecture for several cases, e.g., $r=2$ or $n-2$; $n\leq7$; $n=2r-1$; $h\leq2+1/\,(n-r)$ and $n\leq2r$. The last result uses the Erdős–Ko–Rado theorem. We establish a connection between Ruckle's conjecture and the Hoeffding problem of bounding tail probabilities of sums of random variables.

Existence of perfect equilibria in a class of multigenerational stochastic games of capital accumulation

In this paper we introduce a model of multigenerational stochastic games of capital accumulation where each generation consists of m different players. The main objective is to prove the existence of a perfect stationary equilibrium in an infinite horizon game. A suitable change in the terminology used in this paper provides (in the case of perfect altruism between generations) a new Nash equilibrium theorem for standard stochastic games with uncountable state space.

Equilibrium in a dynamic game of capital accumulation with the overtaking criterion

A stationary overtaking equilibrium is constructed for a class of discrete-time games of capital accumulation. A verification of the equilibrium properties is made using some functional characterization of the overtaking optimality in dynamic programming.

The envelope theorem for locally differentiable Nash equilibria of finite horizon differential games

Envelope theorems are established for a ubiquitous class of finite horizon differential games. The theorems cover open-loop and feedback information patterns in which the corresponding Nash equilibria are locally differentiable with respect to the parameters of the game. Their relationship with extant envelope results is discussed and an application of them to a generalized capital accumulation game is provided. An important implication of the theorems is that, in general, the archetypal economic interpretation of the costate vector, namely, as the shadow value of the state vector along the Nash equilibrium, is valid for feedback Nash equilibria, but not for open-loop Nash equilibria.

Capital accumulation games with a non-concave production function

We consider an economy with a single capital stock that two agents strategically exploit by choosing a consumption profile over an infinite time horizon. We analyze two different games and their corresponding equilibria. In one game firms are able to pre-commit and choose simple time functions as their strategies. In the other game agents are assumed to employ Markov strategies. It turns out that in the case of pre-commitment there exists a threshold level of the capital stock such that if the initial stock is above this threshold, equilibrium consumption converges to the efficient steady state while if the initial condition is below it, the capital stock converges to zero. In case of the Markov equilibrium there exists a unique interior steady state that is globally stable.

Construction of Nash equilibria in symmetric stochastic games of capital accumulation

A large class of nonzero-sum symmetric stochastic games of capital accumulation/resource extraction is considered. An iterative method leading to a Nash equilibrium in the infinite horizon game with the discounted evaluation is studied.

Japan and the Changing Regime of Accumulation: A World-System Study of Japan’s Trajectory From Miracle to Debacle

Japan’s trajectory under globalization is critically reviewed using the world-system perspective and the methodology of historical sociology. The Japanese miracle in the post-war period was a result of interplay between world-systemic opportunities and internal and regional institutional transformation. Japanese success invited US policy changes, ending the growth regime of accumulation in which state-led national economic development was pursued with distributional concessions given to workers. It is argued that misguided policies based on incorrect economic theories under the strong yen, pushed by the US since 1985, prepared a bubble that then burst. The institutions that had provided the Japanese miracle became the source of problems as Japan entered the debacle period in the 1990s. The Japanese debacle was part of the phenomenon of a ‘prosperous US and the debacle of the rest.’ This development was a result of the change in the regime of accumulation from a growth regime to a distribution regime where the rentier class took control of distribution and the project of national economic development was replaced by the monopolistic competition of global corporations. For Japan, both traditionalism and neo-liberalism are dysfunctional. In the short run, Japan as a society needs to focus on survival and the maintenance of people’s living standards under the new rules of the accumulation game imposed by the US. In the medium run, Japan needs to challenge US dollar hegemony ushered in by the new rules. In the long run, the Japanese need to examine whether they should keep engaging in the game of capitalist accumulation.

On the endogeneity of Cournot–Nash and Stackelberg equilibria: games of accumulation

We characterize equilibria of games with two properties: (i) Agents have the opportunity to adjust their strategic variable after their initial choices and before payoffs occur; but (ii) they can only add to their initial amounts. The equilibrium set consists of just the Cournot–Nash outcome, one or both Stackelberg outcomes, or a continuum of points including the Cournot–Nash outcome and one or both Stackelberg outcomes. A simple theorem that uses agents’ standard one-period reaction functions and the one-period Cournot–Nash and Stackelberg equilibria delineates the equilibrium set. Applications include contribution, oligopoly, and rent-seeking games.

SPECIAL CLASSES OF QUIET ACCUMULATION GAMES

In an accumulation game, a hider places objects at locations, and a seeker examines these locations. If the seeker discovers an object then he/she confiscates it. The goal of the hider is to accumulate a certain number of objects before a given time, and the goal of the seeker is to prevent this. In this paper we discuss the quiet accumulation game in which the hider is informed of the location searched on a turn only if the seeker finds an object there. We solve the case where the number of steps is 3 and the goal of the hider is to accumulate 2 objects, and the case where the number of steps is equal to the hider's goal.

COMPLEMENTARITY, SUBSTITUTABILITY AND STRATEGIC ACCUMULATION OF CAPITAL

The preemptive role of capital is analyzed in a class of differential games of capital accumulation with reversible investment for two symmetric players. Two dynamic concepts of interaction are defined: feedback substitutability and feedback complementarity. These concepts are useful for exploring the dynamic properties of the stocks. In particular it is proved that if the equilibrium of the game is characterized by feedback substitutability, the firm with the higher initial condition overshoots his long-run level of capital.

Continuous accumulation games on discrete locations

Continuous accumulation games on discrete locations