Subgame Perfect Equilibria in Continuous Time Investment

Subgame Perfect Equilibrium

Cross-Border Resource Management

Rongxing Guo , in Cross-Border Resource Management (Third Edition), 2018

5.3.3 Subgame Perfect Equilibrium

In game theory, a subgame perfect equilibrium (or subgame perfect Nash equilibrium) is a refinement of a Nash equilibrium used in dynamic games. A strategy profile is a subgame perfect equilibrium if it represents a Nash equilibrium of every subgame of the original game. A common method for determining subgame perfect equilibria in the case of a finite game is backward induction.

Here one first considers the last actions of the game and determines which actions the final mover should take in each possible circumstance to maximize his/her utility. One then supposes that the last actor will do these actions, and considers the second to last actions, again choosing those that maximize that actor's utility. This process continues until one reaches the first move of the game. The strategies which remain are the set of all subgame perfect equilibria for finite-horizon extensive games of perfect information. However, backward induction cannot be applied to games of imperfect or incomplete information because this entails cutting through non-singleton information sets.

Reinhard Selten has proved that any game which can be broken into 'sub-games' containing a sub-set of all the available choices in the main game will have a subgame perfect Nash equilibrium strategy (possibly as a mixed strategy giving non-deterministic sub-game decisions). The subgame-perfect Nash equilibrium is normally deduced by 'backward induction' from the various ultimate outcomes of the game, eliminating branches which would involve any player making a move that is not credible (because it is not optimal) from that node. One game in which the backward induction solution is well known is tic-tac-toe, but in theory even Go has such an optimum strategy for all players (Cited from http://www.nationmaster.com/encyclopedia/Subgame-perfect-equilibrium Accessed on 25.02.14.).

The simplest example of this is the 'Chain Store' paradox (Fig. 5.2). The established store ('incumbent') threatens to fight a price war if the newcomer ('entrant') comes in. There is no reason not to believe this threat. And if the entrant does, he will stay out. There are two Nash equilibria in this matrix, lower left and upper right. But in fact the upper right is stronger than just a static Nash equilibrium. It is what Selten called a 'subgame perfect' equilibrium, because just looking at the last part of the game where the incumbent finds himself one the entrant has entered (the 'sub-game'), it would obviously be irrational for him to follow through on this threat.

Figure 5.2. The 'chain store' paradox.

Specifically, for the incumbent, by fighting, he gets 10 points, and by giving in, he gets 30 (Table 5.5). The sequential or 'extended form' of the game makes this clearer. Of course the incumbent would like his threat to be believed, and with less than full information (about motivations, future games, etc.), it may well be. But if both players have the full information, and it is 'common knowledge' that each is fully informed and rational – then such a threat is absurd and will probably never be made.

Table 5.5. The 'Chain Store' Paradox

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RESOURCES

B. Harstad , M. Liski , in Encyclopedia of Energy, Natural Resource, and Environmental Economics, 2013

A Dynamic Common-Pool Problem

The above simple model can illustrate the forces at play also in simple dynamic common-pool problems. One interpretation of variable x _i is that it measures user i's early extraction level. Suppose there are two stages: i first extracts x _i for the first stage and y _i for the second stage. The total extraction level is z _i =   x _i +   y _i , as before. Suppose consumption equals extraction and generates the utility w ^t (·), t  ε   {1, 2}, where w ^t (·) is assumed to be increasing and concave. Then, user i's utility u(·) over the two stages can be written as:

$u(x_i,z_i)=w^1\left(x_i\right)+w^2\left(z_i-x_i\right)$

In the static or normal-form version of this game, user i will simply set both x _i and z _i at the same time. In that case, it follows from Proposition 1 that:

$w^1\prime \left(x_i\right)=w^2\prime \left(z_i-x_i\right)=v\prime$

Hence, given user i's total extraction z _i , i's extraction in period 1 is efficient. In other words, the intertemporal allocation of the extracted amount is efficient even if the total amount itself is suboptimally large.

However, if the users cannot commit, they will choose the extraction levels for the two states sequentially. The first-stage extraction x _i is chosen, and observed by everyone, before the users choose the second-stage extraction, z _i –x _i . With this dynamic or two-stage extensive-form game, it is natural to limit attention to subgame-perfect equilibria. That is, it is required that the strategies at stage two continue to constitute a Nash equilibrium in the game that is played at that stage, no matter what the first-stage actions turned out to be. Since the x _i s are observed before the z _i s are set, the latter choices are likely going to be functions of the set of x _i choices. That is, z _i   = z _i (x), where x  =   (x ₁,   …   , x _n ).

Definition 3

The strategies ${\left(x_i,z_i\left(x\right)\right)}_{i\epsilon I}$ constitute a subgame-perfect equilibrium if they constitute a Nash equilibrium and if also ${\left(z_i\left(x\right)\right)}_{i\epsilon I}$ constitute a Nash equilibrium at the second stage for every feasible x  =   (x ₁,…, x _n ).

In the two-stage model, the total payoff for i from choosing x _i , given x _{−   i} , is

$u\left(x_i,z_i\left(x_i,x_{-i}\right)\right)+v\left(s_0-{\displaystyle \sum _{j\epsilon I}{z}_j}\left(x_i,x_{-i}\right)\right)$

Player i can thus anticipate how its first-stage action x _i influences extraction not only by i but also by every other j. The authors use superscript d to distinguish the dynamic game from the static. There is a unique subgame-perfect equilibrium.

Proposition 2

Consider the dynamic common-pool problem:

(1)

User i's total extraction z _i ^d is larger than the first-best.

(2)

Given z _i ^d first-period extraction is too large and given by:

[3] $w^{1\prime }\left(x_i\right)=\left(1-\frac{\left(n-1\right)v\prime v″}{nv″+w^2″}\right)v\prime <v\prime$

(3)

Consequently, both x _i and z _i are larger than in the static version of the common-pool problem, while every payoff U _i is smaller:

$z_i^d\ge z_i^s,s^d\le s^s,\text{and}{U}_i^d\le U_i^s$

All inequalities are strict as long as v″   <   0.

Proof

The proof is omitted as it would simply be a special case of the next proof.

In other words, each user extracts more when the other users can observe early extraction and thereafter modify their next extraction rates. Since v is concave, more extraction today induces the other users to extract less tomorrow, since the marginal cost of extracting more, $v^{\prime }\left(s_0-{\Sigma }_I{x}_i\right)$ , is larger if $x_i$ is large. Anticipating this, each user extracts more than in a similar static game.

Note that eqn [3] implies that equilibrium extraction is more aggressive compared to the optimum if n is large and v is very concave compared to the concavity of w ²(·) (i.e., if v′′/w ² ′′ is large). If v were linear, there would be no such distortion, and x _i would be the first-best, given z _i . For any strictly concave v(·), however, the dynamic common-pool problem is strictly worse than the static version of the game. If v(·) were convex, the opposite would be true: user i would then want to extract little at stage one, since this would raise the benefit for the other users to conserve the resource at stage two.

If the resource s is interpreted as clean air and z _i as pollution, then the corollary is simply that each polluter emits more today in order to induce the other users to pollute less in the future. The consequence is more aggregate pollution. If the resource is the stock of fish, each fisherman fishes more than the first-best – not only because he fails to take the static externality into account but also because more extraction today implies that the other fishermen find it optimal to fish less tomorrow. If the resource stock grows between the two stages, the model should be slightly modified but the main results would continue to hold.

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INFORMATION AND BELIEFS IN GAME THEORY

Bernard Walliser , in Philosophy of Information, 2008

6.3 Epistemic justifications of equilibria

An extended reasoning process followed by hyper-intelligent players leads to 'epistemic justifications' of static equilibrium notions [Walliser, 2006]. With the only assumptions of common knowledge of the game structure and of the players' rationality, the relevant equilibrium notion is 'sophisticated equilibrium', obtained by iterated elimination of dominated strategies. With the additional assumption that players play independently, the relevant notion is 'rationalizable equilibrium', where each strategy is a best response to others' strategies, considered as best responses, and so on. With the other additional assumption that the players have a common prior on the state space, the relevant notion is 'correlated equilibrium', where an outside entity, the 'correlator', chooses probabilistically an issue of the game and indicates to each player what it should do. Surprisingly, the two alternative assumptions taken together are not enough to justify a Nash equilibrium. To obtain it, it must moreover (rather heroically) be stated [Aumann and Brandenburger, 1995] that the strategies of the players become shared belief (for two players) or common belief (for more players).

Dynamic equilibrium notions are apparently easier to justify cognitively. For an extensive form game without uncertainty, a 'subgame perfect equilibrium' obtains when common knowledge of rationality applies at each node. The problem is that a player may observe a deviation from the equilibrium path during the play of the game and needs to interpret it. According to a standard result [ Aumann, 1995], a deviation just cannot happen under the main assumption and the subgame perfect equilibrium is perfectly justified. However, when such a deviation is counterfactually considered as possible, a player may wonder what assumption sustaining the equilibrium is not satisfied [Reny, 1993;Binmore, 1997]. The relevant equilibrium notion depends on that precise assumption. For instance, considering the 'trembling hand' assumption (the player may deviate from the intended action with some exogenous probability) preserves the subgame perfect equilibrium [Selten, 1975]. At the opposite, lack of common belief of rationality enlarges the set of possible equilibria.

In both cases, the results were obtained by introducing more and more epistemic logics in classical game theory. In that respect, the state space of the system, which already includes the nature's state, has to be extended to the players' strategies. Conversely, the selection of some equilibrium state in case of multiplicity is treated in a more informal way. Some 'selection principles' are exhibited which are more or less homogenous with the former 'implementation principles'. A first path is to consider that some states are 'culturally' salient hence are conjointly selected as 'focal points' [Schelling, I960]. But salience refers to cultural traits which are not included in the game structure and are essentially context-dependent. A second path is to assume that some selection rules are grounded on various properties of equilibrium states (Pareto optimality, simplicity, symmetry) and act as common 'conventions' among players. But the origin of such conventions is not made explicit and may be history-dependent.

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ENVIRONMENT

M. Greaker , in Encyclopedia of Energy, Natural Resource, and Environmental Economics, 2013

Basic Setup

This model includes two countries: one domestic and the other foreign. There is one nationally owned firm in each of the countries. Both firms pollute, and the governments regulate emissions by some kind of environmental policy. The firms export all their output to a third market. Moreover, they compete by either choosing output levels or product prices, and they take the environmental policies in the two countries as given. It is assumed that pollution is not transboundary and that the environmental performance of the industries has no effect on demand from the third market. Products can be either perfect or imperfect substitutes, although for price competition most contributions assume the latter.

The model is solved as a two-stage game. In the first stage, the domestic country resolves its environmental policy. Environmental policy could be an emission tax or an emission standard. In the second stage, both firms simultaneously decide their abatement levels and their output (price). In order to find the subgame perfect equilibrium in the game, we solve the game by backward induction.

Environmental policy will influence the Nash equilibrium in the second stage of the game. It is then of crucial interest how environmental policy affects the costs of the domestic firm. Normally, it is assumed that both total costs and marginal costs increase. In Figure 1 , given this assumption, we show how environmental policy changes the Nash equilibrium depending on the type of competition.

Figure 1. Cournot versus Bertrand competition.

In both figures, the solid lines represent the best-response curves of the firms without any environmental policy. Moreover, the stippled line represents the best-response curve of the domestic firm after the domestic government has introduced its environmental policy. In both figures, the Nash equilibrium changes from A to B.

In the figure to the left, each firm chooses an output quantity, that is, we have Cournot competition. Note that both best-response curves are downward sloping, and hence, the strategic variables are substitutes. After the introduction of environmental policy, the best-response curve of the domestic firm shifts toward the left. That is, for every level of output of the foreign firm, the domestic firm will supply less. When marginal cost for one of the firms increases because of the introduction of environmental policy, marginal cost will exceed marginal revenue, and the firm will reduce its output. Since the strategic variables are substitutes, the domestic firm produces less and the foreign firm produces more in the new Nash equilibrium.

With Bertrand competition, the situation is different. Now, both best-response curves are upward sloping, and the strategic variables are complements. As seen in the figure to the right, the best-response curve of the domestic firm shifts toward the right, that is, for every price of the foreign firm, the domestic firm will set a higher price. When marginal cost for one of the firms increases, marginal cost will exceed marginal revenue, and the firm will increase its price. Since the strategic variables are complements, both firms have increased their price in the new Nash equilibrium.

When setting environmental policy, it is assumed that the domestic government maximizes net surplus from the market in question. Since there is no domestic consumption in the third-market model, net surplus is simply the sum of the profit of the domestic firm and the domestic environmental costs. The latter are of course negative. In the competitive case, optimal environmental policy is then found as the level of regulation that equalizes marginal abatement costs and MED costs. This is shown in Figure 2 for an emission tax denoted t*, which results in the emission level ε*.

Figure 2. Optimal environmental policies in the third-market model.

However, with imperfect competition, there is also a strategic effect of environmental policy. In the Cournot case, a higher emission tax leads the domestic firm to contract its output. This induces the foreign firm to expand its output, which hampers the profit of the domestic firm since the market price falls. Hence, in the Cournot case, the strategic effect makes it optimal for the government to set the emission tax such that the marginal abatement cost falls short of MED as illustrated by t ^C in Figure 2 . Thus, we have eco-dumping.

With Bertrand competition, the sign of the strategic effect is reversed. Now, a higher emission tax leads the domestic firm to increase its price, which in turn induces the foreign firm to increase its price. This benefits the domestic firm since demand is shifted toward its products. Hence, in the Bertrand case, the strategic effect makes it optimal for the government to set the emission tax such that the marginal abatement cost exceeds MED as illustrated by t ^B in Figure 2 .

Twenty years ago, Michael Porter, Professor at Harvard University, published a one-page essay in Scientific American, entitled 'America's green strategy.' There he argued that stricter environmental regulations could enhance business performance through innovations and efficiency gains. Clearly, there are several ways to interpret the Porter hypothesis. In the SEP literature, the hypothesis is understood as an environmental policy that is stringent enough to make marginal abatement cost exceed marginal environmental damage as in the Bertrand case above. In line with Porter's original paper, we will coin this 'green strategy'. On the other hand, in the Bertrand case, the green strategy will likely reduce the market share of the domestic firm since the domestic firm will in most models of Bertrand competition increase its price by more than the foreign firm.

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Towards High Performance Manufacturing

H. Wu , M. Parlar , in International Journal of Production Economics, 2011

Abstract

In most existing literature in supply chain management it is assumed that the players possess complete information about the game, i.e., the players' payoff (objective) functions are assumed to be common knowledge. For static and dynamic games with complete information, the Nash equilibrium and subgame perfect equilibrium are the standard solution concepts, respectively. For static and dynamic games with incomplete information, the Bayesian Nash equilibrium and perfect Bayesian equilibrium, respectively, are used as solution concepts. After presenting a brief review of the static and dynamic games under complete information, the application of these two games in inventory management is illustrated by using a single-period stochastic inventory problem with two competing newsvendors. Next, we illustrate the Bayesian Nash and perfect Bayesian equilibrium solution concepts for the static and dynamic games under incomplete information with two competing newsvendors. The expository nature of our paper may help researchers in inventory/supply chain management gain easy access to the complicated notions related to the games played under incomplete information.

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Performance measures and metrics in outsourcing decisions: A review for research and applications

Angappa Gunasekaran , ... Thanos Papadopoulos , in International Journal of Production Economics, 2015

3.1.1 Pre-outsourcing stage decisions

Decision-making at this stage is usually based on whether outsourcing, being a major strategic decision, will permit organizations to develop and leverage the capabilities required to compete in today׳s global business environment. McIvor (2008) presented a practical framework that managers can use in the pre-outsourcing phase to identify suitable outsourcing strategies for their processes. The framework provides a number of important insights for managers who want to develop and implement outsourcing strategies in their business processes: (i) see beyond the indicators of poor process performance; (ii) analyze the processes and interdependencies; (iii) understand the potential prior to outsourcing the process; and (iv) use the contract and relationship strategy as complementary. The sourcing strategies presented considering two influential theories, viz., transaction cost economics (TCE) and the resource-based view (RBV) in four quadrants are both useful for developing analytical models for optimization of outsourcing activities.

Since companies are increasingly opting to outsource, supplier selection has become a major strategic decision (Kannan and Tan, 2003; Huang and Keskar, 2007). The following performance measures have been suggested for supplier selection: product-related measures (reliability, responsiveness and flexibility), supplier-related measures (cost and financial status, assets and infrastructure) and society-related measures (safety and environmental). Also, at the integration level, it is necessary to look at the level of integration that exists between the original equipment manufacturer (OEM) and suppliers, to look at operational integration and to examine strategic partnerships. Metters (2008) presented a comprehensive typology to apply to offshore outsourcing decisions. He claims that while offshoring is a possible option, a certain proportion of service processes will remain in high-wage countries. Li et al. (2009) developed a dynamic model to select contract suppliers under conditions of price and demand uncertainties.

Other studies have used game theory for pre-outsourcing decisions. For instance, Li et al. (2009) developed two dynamic three-game models based on production costs and scope economics to study the multi-client outsourcing (MCO) phenomenon that is one vendor vs. multiple clients. They applied the concept of sub-game perfect equilibrium to analyze the three-game models by backward induction and the criterion used is the vendor׳s profit maximization. Tjader et al. (2010) present a multi-criteria decision-making methodology using Analytical Network Process (ANP) to develop an evaluation framework from the perspective of decision-makers, stakeholders, and influential groups. They evaluated four policy options with respect to approximately 50 economic, political, technological and other factors. Araz et al. (2007) developed a fuzzy goal programming model for a textile company that could be used as an external vendor evaluation and management system. This is based on a well-known methodology called PROMETHEE (Multi-Criteria Decision aid Method and Fuzzy Goal Programming). They classified supplier selection methods into five categories: categorical approaches, artificial intelligence (AI), rating/linear weighting models, mathematical programming models, and integrated approaches.

Liou et al. (2011) proposed a hybrid multiple-criteria, decision-making model (MCDM) for selecting an outsourcing service provider combining decision-making trial and evaluation laboratory (DEMATEL), to establish the relations-structure model in evaluation problem, with fuzzy preference programming (to decide upon the pairwise comparisons from imprecise judgement), and ANP (to determine criteria weights with dependence and feedback) methods. Feng et al. (2011) developed a decision method for selecting a pool of suppliers for the provision of different service process/product elements. It employed collaborative utility between partner firms for supplier selection. Multi-objective 0-1 programming was used to select desired suppliers. They proved that the model was NP-hard and developed a multi-objective algorithm based on Tabu search to solve the problem. Isklar et al. (2007) proposed an integrated intelligent decision support model for effective supplier selection problems. This model combines different techniques in order to take advantage of the reasoning power of these techniques.

Bhattacharya et al. (2003) suggest a risk management perspective for IT/IS outsourcing research. Business process outsourcing may appear to be just IT/IS outsourcing, but in practice it represents a global phenomenon. Some of the major criteria used in the selection of ERP suppliers are: interfaces with other systems, price, market position, corporate image and international orientation. They also included some additional criteria, namely: customer service, reliability, availability, scalability, integration, financial factors, security and service-level management.

At the pre-outsourcing stage, scholars have used TCE and RBV to explain outsourcing complexities. But, according to McIvor (2009), "neither TCE nor the resource-based view (RBV) alone can fully explain the complexities of outsourcing". Instead, he proposed a prescriptive framework for evaluating outsourcing, integrating TCE and RBV. Transaction costs are the costs of activities after the product and service are ready to be exchanged between suppliers and clients or customers. This focuses on how much effort and cost is required for the buyer and seller to complete an economic exchange or transaction (Coase, 1937; Williamson, 1975, 2008; Oliva and Watson, 2011). According to Grover and Malhotra (2003), the three most important factors that influence the outcome of TCE are asset specificity, uncertainty and governance mechanisms. Uncertainty is related to performance evaluation, information asymmetry problems, the environment, technology and demand volume and variety. The governance mechanisms depend upon the hierarchies within firms, cooperative behaviour in the buyer-supplier relationship, and increased frequency of communication and control. The RBV includes the firm׳s assets, organizational processes, and information technology and knowledge (Barney, 1991). Also, the importance of operations management concepts such as performance management, operations strategy, business improvement and process redesign for the study of outsourcing is highlighted. Espino-Rodriguez and Padron-Robaina (2006) reviewed the literature on RBV and outsourcing, and illustrate the differences between TCE and RBV in that although both contribute to the definition of organizational boundaries (that is, outsourcing vs. internalization), they differ in that: (i) TCE suggests that when activities based on specific resources are outsourced, the performance of the firm can be negatively affected, because of the increased risk stemming from opportunistic behaviour. However, from a RBV perspective, the decision to outsource depends on the "extent to which activities permit the exploitation of different knowledge, capabilities and routines within the organization" (p. 55); (ii) TCE does not recognize the importance for a firm to focus on its core competencies and safeguard its strategic resources (Prahalad and Hamel, 1990). Therefore, it does not focus on analyzing the capabilities of the organization, and its potential partners or suppliers during decision-making. However, Espino-Rodriguez and Padron-Robaina (2006) suggest that RBV and TCE are not opposed, but are complementary in analyzing outsourcing strategies.

Finally, a variety of mathematical models have been reported for modelling strategic decision-making, including a game-theoretic approach. For example, Xiao et al. (2007) studied production and outsourcing decisions made by two manufacturers that produced partially substitutable products; and they played a strategic game involving quality competition. Both manufacturers outsourced key components to the same upstream supplier. As a result, their products became more substitutable due to the increased availability of the products.

The PMMs for the during-outsourcing stage are discussed in the next section.

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