THE AUSTRALIAN NATIONAL UNIVERSITY

WORKING PAPERS IN AND ECONOMETRICS

Network Externalities and Institutional Adaptation1

Nhat Le

School of Economics Faculty of Economics and Commerce

Working Paper No. 427

November 2002

ISBN: 086831 427 7

1 Faculty of Economics, Australian National University, Canberra, ACT 0200, Australia. [email protected], phone: (61 2) 6125 54442, fax: (61 2) 6125 3700. Earlier versions of this paper were distributed under the title: “Multiple Game Linkages in Evolutionary Framework.” I would like to thank Steven Tadelis for his guidance. I wish to thank Peyton Young for his critical comments on several drafts. Thanks are also due to , Douglas Bernheim, Marcelo Clerci-Arias, Avner Grief, Jonathan Levin, John McMillan, Rod Tyers and the participants of the comparative institutional analysis seminar at , the theory workshop at the Australian National University. Financial supports from ACLS and RSPAS are greatly acknowledged.

Abstract

This paper presents a dynamic framework that explains how a set of institutions emerges when players extrapolate across multiple games. It explores the existence of a fundamental circularity whereby the high convention in one game reinforces the high convention in others, and vice versa, such that one possible outcome is a socially advantageous regime combining the high conventions. Likewise, the low conventions also reinforce one another to form a socially disadvantageous regime. A convention in one game reinforces the corresponding convention in the other game by altering the payoff structure in favor of the latter. The payoff structures of the games explored here are driven by the competition between the two alternative regimes. In the long run, however, the regime that adapts better into its milieu, or equivalently, is more firmly rooted in past adaptation, will overcome the alternative. This dominant regime will determine the asymptotic outcome in all games.

JEL Classification Numbers: C7, D7, D8, O0

Keywords: Adaptation, Complementary institutions, Coordination games, Risk dominant strategies, Selection of conventions, Stochastically stable states.

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I. Introduction

When inquiring into the causes, nature, and implications of institutional change historians and development economists have long sought to consider the economic system as a coherent whole. Yet the nature of institutional change in the presence of complementarities is still not fully understood by researchers and policy makers alike. When designing policies in response to changes in conditions, policy makers still lack theoretical models that deal with institutional complementarities. There are some exceptions however. Li

(1999) point out that the economic system is a complex set, which has many layers. An ex ante comprehensive policy, which aims to alter every aspect of the existing system altogether, may lose its complementary ex post when some parts of that system respond to the policy much more slowly than the others. From their point of view, it is reasonable to accept the view of Ogburn (1950) that the actual institutions at any time represent adaptation in part to past conditions. This problem is reinforced by network externalities between different institutions (Arthur, 1989; David, 1993). Thus, from a short-term perspective, a key property of the system is its inertia; the expected waiting time until the system transiting from one regime to another may be enormous.

In the corresponding literature on development, some related issues have also been discussed. Stigler

(1951) and Jacobs (1969) indicate that many ad hoc measures to transplant new organizations, rules, or technologies into a place that lacks a network of auxiliary institutions often meet with failure. Echoing them, Ciccone and Matsuyama (1996) note that there is a fundamental circularity between the choice of technologies by firms and the variety of intermediate inputs available, which include legal support, accounting, and financial services. Their analysis highlights linkages in the selection of conventions across markets. But they did not address this issue formally. The objective of this paper is to provide such an analysis. The focus is on reciprocal relationships in the selection of convention across spheres of human

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interactions; how a particular set of complementary institutions comes into being; and why some institutions persist even when they are no longer optimal.

This exercise requires us to specify the concept of selection of institutions. In the economists’ view, an institution is an equilibrium outcome of a game, which has multiple equilibria (Sugdent, 1989; Hurwicz,

1996). The selection of an institution, therefore, boils down to the question why one particular equilibrium outcome of the coordination game is chosen over the others. The most widely accepted selection concept is that of perfect Baysian equilibrium (see, for example, Myerson, 1991; Fudenberg and Tirol, 1991). It suggests that the system of belief and rational behavior together pin down what equilibrium of the game can be selected as a convention. There are some criticisms, however. First, the system of beliefs is often given exogenously. For instance, in the history literature, individuals’ preference sometime is determined either by rule makers (North, 1990) or by cultural patterns (Weber, 1951; Grief, 1994). Second, like most models in neoclassical economics, in standard game theory, individuals or players are assumed to be highly rational. This is rather an extravagant description of human behavior (Young, Ch1., 1998). In reality, people often have very limited understanding about the world around them, including the preferences of their partners.

The learning theory has filled this gap by making selection of equilibrium endogenous rather than exogenous. In essence, the system of beliefs is formed in an adaptive, low rationality environment. Perhaps, the most prominent step in this approach is “trembling – hand perfection” by Selten (1975), which assumes people to behave “less” rationally by introducing the possibility of mistakes. A convention then is expected to emerge in the long run only if it is robust to mistakes or errors. Interestingly, Selten’s notion of convention selection is closely related to what Hayek, Menger, and other members of Austrian school called “spontaneous order”. But this notion may also have contributed to the development of two important concepts: the “risk dominant strategy” of Harsanyi and Selten (1988) and “adaptive play” by Foster and

Young, (1990). In essence, there are some strategies that are least risky in the view of players. In the terminology of Harsanyi and Selten, these are risk dominant strategies.

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If learning has any implication on the formation of expectations, the game should converge to a risk dominant strategy (Fudenberg and Levin, Ch. 1, 1998). Young (1993, 1998) has formalized this idea. In his framework, a player forms her expectation about the opponents’ action by sampling from her memory of previous plays. She then takes the best reply to that belief. But with some small chance, she makes a mistake and chooses a strategy other than what is suggested by her belief. Such mutations or errors play a crucial role in promoting long-term change. In particular, the theory suggests that, in the presence of random shocks or mutations, some conventions are inherently more stable than others. Young called them

“stochastically stable states”, which in fact correspond one to one with risk dominant strategies in a 2× 2 coordination game (Young, Ch. 4, 1998). Over the long run these institutions occur with higher frequency than others. A corollary is that the selection process is ergodic, not path dependent.

Young’s adaptive play is very useful for the purpose of this paper. However, since players are not allowed to extrapolate across games, the selection process offers an idiosyncratic rather than a perspective with institutional complementarities. Further, given network externalities, the selection of conventions is influenced by past adaptations (Arrow, 1997). History, therefore, matters (North, 1973; Grief, 1994). This feature is then re-emphasized by Young himself (Young, Ch. 1, 1998). But because players interact only in a single game, the structure of institutional linkages is hidden or absent. Thus, one learns little about the effect of past adaptation on the subsequent trajectory of institutional development. In this paper, Young’s work is built upon by constructing a formal model to analyze the selection of a set of complementary institutions to coordinate individuals’ behavior. Since the focus is on reciprocal relationships in the selection of equilibrium outcome across games, a set of corresponding conventions is sought, each of which reinforces another to emerge simultaneously. It is assumed that the high equilibrium in one game reinforces the high equilibrium in the other game and vice versa, to form the “high regime”. The two low equilibriums reinforce one another to form the “low regime”. As suggested by Grief (1994) and North (1995), a convention in one game reinforces the corresponding convention in the other game by altering the payoff structure in favor of the latter. We then allow the payoff structures in both games to be driven by competition between the two alternative regimes. In the long run, however, the regime that adapts better

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into its milieu, or equivalently, is more firmly rooted in past adaptations, will overcome the alternative.

This dominant regime then determines the asymptotic outcomes in both games.

As already suggested, the notion that a set of conventions reinforcing each other to form a regime is scarcely new idea in economics. But here, we address this issue formally, using the evolutionary approach.

The rest of the paper is organized as follows: Section II presents the essential ideas to be later formalized, using a field study by Baker (1998). Section III provides a theoretical framework. Section IV concludes the paper.

II. Selection of Conventions through Examples

To pave the way for later discussions, let us consider Baker’s study about rural Africa. Historically,

Africans have faced harsh and unpredictable conditions. Various types of insurance have arisen in response to this hardship. Baker (1998) argues that, at least partially, the reciprocal relationship between these types of conventions has created a hostile environment for technological change. Baker is most concerned about two particular types of conventions: one is “retirement income” and the other is “ex-post community risk sharing”. In this section, I reformulate Baker’s model and put it into an evolutionary framework. This highlights some characteristics of the selection process in the presence of complementarities.

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II. 1 Retirement Income

Let us assume that farmers live for two periods. A peasant who born at time t works only in the first period of life. In the second period, he becomes old. He must use his savings to finance the consumption.

Alternatively, he may use the experience he has gained through working in the same field for years and hire a young person to work. We may call the former a financing relationship and the latter a multi-generational relationship.

Let us first consider the multi-generational convention. Let π (e, k) denote the total farm profit, which is

the function of experience e and technology k (e = 1,2 and k = ko , k n , according to the aging period and the old or new technology, respectively). Labor is ignored since it is always unity.

The young peasant receives a reservation wage π (1, k) paid by his head – the old farmer. The old farmer collects the residual income π (2,k) −π (1,k) from the young’s work. In such a farm, a convention seems to emerge: the head assumes all the bargaining power and the farm has a strong tendency to use backward technologies in production. Let us provide a rational for this fact.

Baker assumes that the new technology, k n , is superior: π (e,kn ) > π (e,ko ) but that it erodes the old farmer’s experiential advantage:

π (2, k n ) − π (1, k n ) < π (2, k o ) − π (1, k o ) (1)

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Intuitively, this experience is gained over time. It would become valueless if new technologies were adopted. Therefore, it is not in the old farmer’s interest to adopt the new technology. Since the farm belongs to him, he would not authorize any action that may work against his power. Anticipating this behavior, the young peasant will not to get involved with the new technology. Thus, the farm may stagnate

in the old technology ko . In payoffs, the head gains π (2,ko ) −π (1,ko ) , while the subordinate receives

π (1,ko ) .

The old farmer, however, may sell his farm, whose value amounts to his saving. He can lend this money to someone else and collect interest plus the principal as means to finance his retirement. The young peasant also has another option: instead of working for someone, he can borrow money to start up his own farm.

Given the new technology is superior, the young farmer should choose to invest in the new technology k n .

Notice that experience no longer matters; it is convenient to explicitly provoke the production function

' f (kn ) . The young entrepreneur pays back the old farmer an amount: kn f (kn ) = rs , and keeps the

' 1 residual π (kn ) = f (kn ) − kn f (kn ) , where s is the old farmer’s saving, and r is the real interest rate .

We assume this financing relationship is socially optimal, which implies:

' sr = kn f (kn ) > π (2,ko ) −π (1,ko ) (2)

Rationally, an entrepreneurial young farmer will search for an old farmer who prefers to use saving to finance his retirement consumption. And because the new technology yields the highest return, that old farmer will also prefer to lend his money to a young entrepreneur, who invests in the new technology. This makes the financing relationship incentive compatible.

1 Note that π (k n ) = π (1, k n ) , but we drop the variable e, since it always equals one. This case is thus distinguished from that where the financing relationship is not involved.

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The feature we want to emphasize here is that, although most of people will adhere to this convention, an old farmer might miscalculate and lend his saving to the wrong partner, who instead of investing in the new technology dumps the funds in the old one. If it happens, the expected returns for the lender are low. First, investing in the old technology will yield a low return, which can be seen from (2). Second, the inexperienced young farmer, who works alone in the field, may face a great risk of not being able to handle crises, such as bad weather and the like, making the prospective return even lower. Third, in any traditional society, if the project for which the loan was used performs so badly that the borrower may default, there are some norms that limit the lender’s recollection of funds yet leaving the borrower with less that the

minimum income, π (1,ko ) [see Hoff and Stiglitz, 1990]. Subsequently, the lender can expect to gain β ,

which may be lower than π (2,ko ) −π (1,ko ) .

Alternatively, when working for the old farmer, a young one who should follow the head’s instructions to use the old technology may still get involved with the new one. Given the inequality (1), that action will cause a conflict of interests, resulting in some welfare loss, α , that the young misfit must incur (obviously, we still assume that the head has the authority to impose punishment). In payoffs, the head of the farm

gains π (2,kn ) −π (1,kn ) , but his subordinate gains only π (1,kn ) −α . If the cost of punishment, α ,

is sufficiently large, we have: π (1,kn ) −α < π (1,ko ) . The game then can be summarized as below

[Insert Table 1 here]

The game has multiple equilibria: the financing relationship and the multi-generational one. A question then arises: what convention will be more likely to be observed in the long run? The answer depends on the structure of payoffs. In particular, players will not prefer a strategy that might cause a relatively large loss if misfit is encountered. For example, if the advantage of the new technology over the old one is moderate, while the loss when facing a misfit is substantial ( β is low), then using saving to make income is too risky a strategy for the old farmer. It may not be worthwhile for him to sell his land and lend this lump sum to a

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young one, who may mistakenly dump his money in the old technology. Similarly, choosing the new technology may not be safe for the young farmer if the cost associated with mismatch, α , is sufficiently large. In Harsanyi and Selten’s terminology: expertise - old technology is risk dominant.

By contrast, if the advantage of the new technology over the old one is substantial, and the cost associated with making mistakes, α and β , is relatively low, more people will get involved with technological innovations. Occasionally, these mutations may become sizable enough to tip the system toward the socially optimal convention. Of course, the system can go in the reverse direction, but it may be too few people who want to give up the new technology to use the backward one. Such mutations would never become sizable in the given context. In Young’s terminology, it is less resistant for the system to transform from the low equilibrium to the high equilibrium, thereby making the latter conventionalized.

To formalize the concept introduced here let us use the concept of adaptive learning by Young. For the

sake of generality, let us consider a 2× 2 game G with the joined strategy sets X = X 1 × X 2 . In each

period, one player is drawn randomly from each of two disjoined classes C1 ,C2 . For example, these are

t the classes of the old and the young farmers. Let xi denote the action taken by the class i player at time t.

t t t The record is the vector x = (x1 , x2 )' ; and the state of the system is characterized by the sequence of the last m plays: h t = (x t−m+1 ,...., x t ) . Let X m denote the set of all states or the set of all histories of length m. At the beginning of period t+1, the current i player chooses his action as follows:

- With probability (1 − ε) , he draws a random sample of size s from the set of actions taken by the

t opponents over the last m periods, and plays the best response to the resulting sample proportions p−i .

- With probability ε , however, he may make an error by choosing an action xi ∈ X i at random, each with equal probability. Such errors correspond to mutations mentioned above.

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This sequence defines a Markov process P m,s,ε on the state space X m consisting of all length-m history.

We call it adaptive play with memory m, sample s, and error rate ε (Young, 1993). A fundamental result is that in the long run, when random shocks, ε , go to zero, such an adaptive play will converge to one of risk dominant conventions, which Young calls stochastically stable states (Young, 1998, Ch. 4).

II.2 Community Risk Sharing

In an adaptive environment, the risk dominant convention will be likely to emerge if time is left to unfold.

In the short run, however, other behavior patterns can coexist. Particularly, in some farm, the owner is a young entrepreneur, who wants to invest in the new technology, while in a nearby farm the owner is an old peasant who prefers to use the old technology to cultivate. It makes the village cultivating system a

“scattered strip system” in which farms with different types of technologies are scattered and mutually intermeshed. The game then should be extended to take into account the interactions between different types of owners for the need of, say, sharing some risks. This new game may have an important implication for the stability of a particular norm in the previous game. Let us then enrich the previous setting by introducing random factors into our analysis.

For the sake of convenience, let us call the old farm owner by Y and the young farm owner by X. As

~ already shown, the player Y always prefers to use the old technology to achieve a profit y = π (2, k o ) , but this time, it is random. Similarly, the player X always chooses the new technology to achieve a random

~ profit x = π (kn ) . Further, let us assume that:

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2 2 µ y < µ x , σ y = σ x (3)

2 Where, µ x and σ x denote the expected value E(x) and the variance Var(x) of the random variable x , respectively.

The two owners then seek a risk-sharing agreement. We are only interested in the community’s response to idiosyncratic risk since communal insurance cannot deal with community-wide risk. This implies players’ profits are independent. The purpose of ex post risk sharing is to minimize the variance of profits.

As is well known, if the ex post risk sharing schema is linear in the difference of the realized profits, the

1 transfer payment from player Y to player X will take the form: τ (x, y) = 2 [(y − x) − (µ y − µ x )] . The implicit assumption here is that no one finds bearing risk more or less onerous than does the other. In other words, they both have the same degree of absolute risk aversion.

After transfers, the expected profit is the same for the old farmer, but the cost of bearing risks is reduced:

E(y −τ (x, y)) = E(y) = µ y

1 2 2 1 2 Var(y − τ (x, y)) = 4 (σ y + σ x ) = 2 σ y (4)

Similar results apply for player X.

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To make the picture more realistic, Baker assumes that Y cannot accurately assess X’s profits (since his

technology of production is new). Specifically, let xa be the old agent’s assessment, which is a random variable distributed with parameters (x,σ 2 ) , where x = E(x  x ), σ 2 = Var(x  x) . We then xa a xa a have:

E(xa ) = E[E(xa  x)] = E(x)

Var(x ) = E[Var(x  x)] +Var[E(x  x)] = σ 2 + σ 2 (5) a a a xa x

Equations in (5) mean that information problems add more uncertainty into the new technology. Player Y now perceives that his young partner – player X – is involved in a riskier project than is he. The more severe is the information problems, σ 2 , the more likely the player Y will demand the player X to pay xa him an additional transfer to make him as well off as he was before. An implicit assumption here is that bargaining power lies with the former. Such an additional burden, τ (σ 2 ) , depends on the actual size of xa the “noise” in player Y’s assessment of player X’s profit, σ 2 . Therefore, when adding up, the cost of risk xa born by X could be higher than the level justified by his risk tolerance. In this case player X’s expected utility would be eroded, perhaps enough for him to decide not to invest in the new technology. This situation clearly happens when the expected value of investment is lower than the threshold value,

π (1,ko ) , that X can receive for sure when being employed by an old farmer. The game can be summarized as below:

[Insert Table 2 here]

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Where p is the cost of bearing risks, which is the same for both X and Y when they get involved in a risk sharing agreement without information asymmetry.

Now let (1,1) denote the equilibrium in which player X invest in the new technology and (2,2) denote the other equilibrium. As already said, if the cost caused by information asymmetry is too large so that:

µ − p −τ (σ 2 ) < π (1,k ) , then “invest” is too risky an action for player X to take. Put differently, x xa o equilibrium (1,1) is not risk dominant. The inverse holds, if µ − p −τ (σ 2 ) > π (1,k ) . x xa o

To formalize the principle that applies here, it is useful to think of the risk factor (Young, 1998). For the

sake of generality, let aij ,bij ,i, j = 1,2 denote the payoffs of the row and column player, respectively.

The game has pure Nash equilibria (1,1) and (2,2) : aii > a ji , bii > bij ,i, j = 1,2 . We define the risk factor of equilibrium (i,i) , where i = 1,2 to be the smallest probability p such that if one player believes the other is going to play action i with probability strictly greater than p then i is the optimal action to

take. For instance, let us consider equilibrium (1,1) . Denote pc (and pr ) the smallest such probability for the row player (and the column one, respectively) to take action 1. The risk factor for equilibrium (1,1) is

then pc ∆pr = min( pc , pr ) and for equilibrium (2,2) , that factor is (1− pc )∆(1− pr ) . A risk dominant equilibrium is the one whose risk factor is lowest. Thus, equilibrium (2,2) is risk dominant if and

only if pc ∆pr > (1− pc )∆(1− pr ) . By some manipulation, this condition is equivalent to

(a11 − a21 ).( b11 − b12 ) ≤ (a22 − a12 )(b22 − b21 ) (6)

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If we apply (6) to the above problem, we see that equilibrium (2,2) is risk dominant if and only if:

µ − p −τ (σ 2 ) ≤ π (1,k ) (7) x xa o

II.3 Reciprocal Relationships

Does the evolutionary setting presented here help us to understand the lack of technological change in rural

Africa? More specifically, what could cause the inferior equilibrium (2,2) in the retirement-income game to be chosen? The explanation lies in reciprocal relationships that make this convention firmly rooted in specific institutional environments. Recall that the need for earning retirement income underlines the old generation’s choice of traditional technology. If by chance events the inferior technology becomes popular, it strengthens the older generation’s power in the village. If a young innovator adopts the new technology and increases his saving, the village heads might think that his harvest has been underestimated

(σ 2 becomes larger). The heads then demand a higher transfer τ (σ 2 ) from him. Expecting that, the a xa young must internalize this cost in his investment decision. But then the expected return from an investment in the new technology is eroded, perhaps enough to make the young decide not to undertake it.

The circularity between the choice of technology and the pattern of risk sharing then has been established.

It works in a way to alter the payoff structures of both games in favor the “multi-generational” convention.

In other words, such a reciprocal relationship between the two games has created an institutional barrier to technological change.

But we can step back further and analyze the impact of other institutions on the selection process in the retirement-income game. Recall that the advantage of the new technology plays a crucial role in

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determining which equilibrium is chosen. Specifically, if either one or both the following slacks are sufficiently large:

kn f '(kn ) = f (kn ) −π (kn ) >> π (2,kn ) −π (1,kn ) , and π (kn ) >> π (1,ko ) (8)

so that:

[kn f '(kn ) − (π (2,kn ) −π (1,kn ))][ π (kn ) − π (1,ko )] ≥

[π (2,ko ) −π (1,ko ) − β ][ (π (1,ko ) −π (1,kn ) +α] (9)

then, instead the financing-relationship equilibrium will become institutionalized.

Notice that the left-hand sides of the first and the second inequalities in (8) are the expected gains of the old lender and the young investor, respectively. The right hand sides in (8) are somewhat different. In the first inequality, we have the old’ s gain from applying the new technology without getting involved with financing relationships. In the second inequality, we have the young’s payoff from using the traditional technology. Thus, there is some reason to believe that the larger are the slacks in (8), or equivalently, the greater the advantage of the new technology, the wider the range of supporting institutions available to innovators.

There is a fundamental circularity between the choice of technology and a set of auxiliary institutions.

These include ownership structure, the organization of the firms and the external network of producer services in which risk-sharing arrangements is only one element. For instance, if a country lacks institutions that reinforce private property rights, it may be too costly for landlords to trade their rights to land, making

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investments in the new technology too costly, regardless of how good this technology may be. In some extreme cases legal and moral enforcements even prevent landlords from selling their land, leaving the aged to rely on altruism and their experience to earn their living. Similarly, in the absence of some rudimentary financial service networks to reduce uncertainty in saving and lending money, farms may be forced to use more primitive modes of production. This in turn implies a limited incentive to elaborate a broader financial service network for investors (Ciccone and Matsuyama, 1996.) On the contrary, if a society were able to develop various institutions to secure property rights to land or to build up alternative patterns of sharing risk that leave the individuals as residual claimants to their own profits; we would expect technological change to occur more easily. The gain from investing in new technologies, if sufficiently large, will induce individuals to experiment with an even a wider set of legal and financial services to protect ownership rights or to facilitate financing relationships.

Thus, there is the circularity between the choice of technology – the selection of equilibrium in the retirement income game - and the emergence of various market-supporting institutions, which stem from other games. These reciprocal relationships may fall into two distinct patterns: either a vicious circle or a virtuous circle. Each builds up its own foundation to strengthen its own specific set of institutions. One associates with the inertia of technology and the other experiences with a rapidly technological change. In the next section, we will develop a formal framework for a rather speculation on institutional adaptation in the existence of network externalities presented here.

III. The Model

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Let us consider an adaptive play by Young (1993, 1998). In this time, however, individuals simultaneously play two 2× 2 coordination games: G1 and G 2 . Without loss, let actions (1,1) be the high equilibrium

j and actions (2,2) be the low equilibrium in both games. Let hi denote the convention associated with equilibrium (i,i) of the game G j ,i =1,2; j =1,2 . We assume for the moment that both G1 and G 2 are

j j j symmetric: ai = aii = bii ,i = 1,2, j = 1,2 , and off-diagonals are zero:

j j aik = bik = 0,∀i ≠ k, j = 1,2 .

We are concerned with the circularity between the selections of equilibrium in different games. We are then interested in situations where a convention in one game reinforces the corresponding convention in the other game and vice versa. The former can reinforce the latter by either increasing the payoffs associated with that equilibrium or decreasing the payoffs associated with the alternative equilibrium. For instance, the low equilibrium in the retirement game reinforces the low equilibrium in the risk-sharing game by subtracting the payoffs associated with the high equilibrium in that game by an amount τ (σ 2 ) . xa

When a pair of conventions has emerged, it forms a coherent whole that we call it a regime. In what follows, we will assume that the high equilibriums in two games G1 and G 2 reinforce one another in a

1 2 virtuous circle to form the superior regime: R1 = {h1 ,h1 } and the two low equilibriums reinforce each

1 2 other in a vicious circle to form the inferior regime: R2 = {h2 ,h2 } .

Assumption 1: The payoffs of the game G j depend on some parameters. That is,

j j j j j j j j j j j j ai = ai (λi ,θ i ),λi ∈ Λ i ,θ i ∈ Θi ,Λ i ∩ Θi = ∅,i, j = 1,2. Further, the parameters λi ∈ Λ i

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j are changing much more slowly than those belonging to Θi , so that the former can be seen as fixed compared to the latter.

j j j The fixed parameter λi reflects the intrinsic value of the convention hi in G . The relatively changeable

j − j part of this convention, θ i , may be altered under the influence of the convention emerged in G . That parameter then is a channel through which the reinforcing linkages work. Specifically,

1 1 Assumption 2: Suppose the convention hi ,i = 1,2 has actually been realized in G . It gradually alters

2 2 2 2 the value of the vector θ = (θ1 ,θ 2 )' in G in according to the following rule:

θ 2 = θ 2 +η ,m = 1,2 m,t m,t0 t

η = λ1 + ρ η + ... + ρ η +ν (10) t i i,1 t−1 i, pi t− pi t

Where, θ 2 denotes the initial value of vectorθ 2 ; parameter λ1 ∈ Λ1 measures the magnitude by which t0 i i

1 2 2 the convention hi comes to influence hm through altering the value of its parameter θ m ,m = 1,2 ; ν t is a white noise.

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We will restrict that: . Under this condition2, when time unfolds, the process (10), ∑ ρi,k < 1 AR( p) k =1, pi

λ1 denoted by θ 2 (λ1 ) , will converge to θ 2 (λ1 ) =θ 2 + i ,m = 1,2 . We then see that if m,t i m i m,t0 (1− ∑ ρi,k ) k =1, pi

1 , the reinforcing linkage is redundant; and if , past “innovations” do not affect the λi = 0 ∑ ρi,k = 0 k =1, pi linkage.

Since we are focusing on a regime, whose institutional elements enforce each other to emerge, we are interested in situations such that:

Assumption 3: The reinforcing linkage must work through a specific set of parameter θ 2 such that:

∂a 2 ∂θ 2 ∂a 2 ∂θ 2 i i , −i −i (11) 2 1 > 0 2 1 < 0 ∂θ i ∂λi ∂θ −i ∂λi

1 1 2 In other words, the convention hi ,i = 1,2 in G reinforces the corresponding convention hi ,i = 1,2 in

G 2 .

j Take into account the reinforcing linkage; the convention hi , j = 1,2 associated with the regime Ri is

j j characterized by the set of parameters: {λi ,θ i ,(ρi,k ,k = 1, pi )}.

2 Needless to say that in order for the process (10) to converge to a finite number, the autocorrelation

λk function ACFk = ;λk = Cov[ηt ,ηt−k ] must eventually taper off to zero. In general, we will require λ0 the roots of the characteristic equation C(z) = 1− ρ z − ρ z 2 − ... − ρ z pi = 0 lie outside the i,1 i,2 i, pi unit circle. Consequently, we will restrict that: . ∑ ρi,k < 1 k =1, pi

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2 2 1 1 Assumption 4: If hi ,i = 1,2 emerges in G , it comes to influence the convention hn in G through its

1 1 2 parameter θ n . In the long run, the processθ n,t (λi ) will converge to

λ2 θ 1 (λ2 ) =θ 1 + i ,n = 1,2 . n i n,t0 (1− ∑ ρi,k ) k =1, pi

∂a1 ∂θ 1 The feedback loop also has to work in a specific set of parameter 1 such that: i i and θ 1 2 > 0 ∂θ i ∂λi

∂a1 ∂θ 1 −i −i or equivalently, 2 in 2 reinforces 1 in 1 . 1 2 < 0 hi ,i = 1,2 G hi ,i = 1,2 G ∂θ −i ∂λi

2 1 1 2 Together, the functions θ (λi ) and θ (λi ) form the “circle” that reflects the reciprocal relationship

1 2 between the two games. When this relationship applies for the high regime R1 = {h1 ,h1 }, which

2 1 1 2 corresponds to the set {θ (λ1 ),θ (λ1 )}, we have the virtuous circle; and for the low regime,

1 2 2 1 1 2 R2 = {h2 ,h2 }, which corresponds to the set {θ (λ2 ),θ (λ2 )}, we have the vicious circle.

j j j j j From hereon, we simply write ai (θ i ) , instead of ai (λi ,θ i ) , unless specified otherwise.

Assumption 5: When there exists the reciprocal relationships between two games, then at period t , the estimated value of the current payoffs in the game G 2 is:

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2 t,ε ,1 2 2 1 t,ε ,1 2 2 1 ai (ε,t) = µ1 ai [θ i,t (λ1 )] + µ 2 ai [θ i,t (λ2 )],i = 1,2 (12)

t,ε ,1 1 Where, µi ,i = 1,2 is the relative frequency of visiting convention hi over the first t periods of the

ε ,1 1 1 adaptive play, P , in G . An analogous expression can be written for ai (ε,t) .

The relation (12) reflects the fact that the prevailing convention in the game G1 constantly and parametrically alters the payoffs associated with certain actions in the game G 2 . It also reflects the competition between the virtuous circle and the vicious circle in driving the payoff structure of G 2 .

In such a dynamic context, in order to define which action s/he should take in G 2 , a player not only needs to form the expectation about what others are going to do, but also s/he needs to update the estimated value

2 1 t,ε ,1 of ai in according to the prevailing convention in G . The higher the frequency µ1 , the more likely parties are persuaded that the virtuous circle dominates the vicious one. From the learning rule (12), the

1 2 convention h1 will affect the assessment about ai more strongly ( i = 1,2) . The inverse holds, when the

t,ε ,1 t,ε ,1 frequency µ 2 is relatively higher than µ1 . An analogous judgment should be drawn from the feedback loop from G 2 to G1 .

2 ε ,1 Next, let us consider the asymptotic behavior of a j (ε,t) . Provided P is a regular perturbed Markov

0,1 t,ε ,1 1 process of P , hence when t → ∞;ε → 0 , the frequency µi → µi , which is a stationary distribution of P 0,1 (Young, 1993). Given the updated process (12), one can see that:

21

Lemma 1: Over the long run when perturbation becomes small,

2 1 2 2 1 1 2 2 1 ai (ε,t) → µ1 ai [θ i (λ1 )] + µ 2 ai [θ i (λ2 )],i = 1,2 (13)

The lemma below completes our analysis with respect to the asymptotic behavior of the variable

2 ai (ε,t),i = 1,2 .

1 1 Lemma 2: For a 2× 2 coordination game G , if the convention hi is the (unique) stochastically stable

t,ε ,1 t,ε ,1 state, then µ i → 1;µ −i → 0 , when t → ∞;ε → 0 .

1 1 Subsequently, if the convention h1 is the stochastically stable state in G , then we have:

2 2 2 1 1 ai (ε,t) → ai [θ i (λ1 )],i = 1,2 ; alternatively, if the convention h2 is the stochastically stable state in

1 2 2 2 1 G , then we have: ai (ε,t) → ai [θ i (λ2 )],i = 1,2 .

Needless to say that in order to complete the circle, we need to derive the analogous explanation for

1 2 2 ai (ε,t),i = 1,2 . In particular, if the convention h1 is the stochastically stable state in G , then

1 1 1 2 2 ai (ε,t) → ai [θ i (λ1 )],i = 1,2 . Alternatively, if the convention h2 is the stochastically stable state in

2 1 1 1 2 G , then we have: ai (ε,t) → ai [θ i (λ2 )],i = 1,2 .

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1 2 Under reinforcing linkages, one should expect that two corresponding conventions, such as {h1 ,h1 } , either emerge together, or alternatively, they have to be redundant together. The following proposition formalizes this intuition.

1 2 Proposition 1: In the presence of complementarities, in the long run, either conventions h1 , and h1 , or

1 2 1 2 alternatively, conventions h2 and h2 together emerges in G and G , respectively. Otherwise, the system is in the state of uncertainty.

1 1 2 Proof: Without loss, suppose under reinforcing linkages, h2 emerges in G , but instead of h2 , the

2 2 1 2 convention h1 emerges in G . Then according to (11), h2 constantly alters the payoffs in G in a way

2 that weakens h1 and vice versa. Subsequently, when time unfolds, we have either

1 1 2 1 1 2 2 2 1 2 2 1 a1 [θ1 (λ1 )] > a2 [θ 2 (λ1 )] or a2 [θ 2 (λ2 )] > a1 [θ1 (λ2 )] . Notice (6), the first case is amount to the

1 1 1 2 fact that h1 now becomes the stochastically stable state of G . In the long run, the convention h1 and h1

1 2 2 2 will be observed more frequently in G and G , respectively. In the second case, h2 overcomes h1 . An

1 2 analogous conclusion is drawn for the pair h2 and h2 . When neither of the above inequalities realizes, we can say that the system exhibits what would appear to be an unwillingness to create expectation-stabilizing regime, with the consequence of considerable uncertainty.

Put aside the case of uncertainty. How can we say about the possibility that one regime will be able to overcome the alternative? As suggested by historical evidences, if one regime adapts to its milieu better than does the alternative, or equivalently, it is rooted more firmly to past adaptations, this regime is more likely to prevail over the latter. Such a situation is not surprised in view of biological evolution. The species

23

that exist are not “superior”. They carry with them the remains of previous adaptations, which have influenced the course of future developments (Arrow, 1998). Let us elaborate this idea in the language of learning process specified in (12).

In the beginning of the adaptive process, there is a steady flow of new information about the payoff structures and the states of the games. Different players learn different parts of that information. Such dispersion of information is caused by the fact that players participate in different stages of the games. It is also because the fact that the payoff structure in each game is a stochastic process [see (10)]. In a longer run, however, if there is any chance at all for a more stable pattern of expectations to be taken shape, one

1 2 regime, say Ri = {hi ,hi }, has to appear with higher frequency than does the alternative regime, R−i ,

t,ε , j t,ε , j according to proposition 1. In other words, the frequency µi is relatively higher than µ −i in

G j , j = 1,2 . By the learning rule (12), individuals’ assessment about the payoff associated with certain actions will be updated accordingly so that now it is less resistant for the system to transit to the convention

j j j hi than to the alternative convention h−i in the game G . This, in turn, implies that when random shocks

j become smaller, the system will be more likely to follow paths that lead toward the convention hi in the

j 1 2 game G , j = 1,2 respectively. Under such a self-reinforcing pattern, the regime Ri = {hi ,hi } will come into being in the long run.

In the above process, there is a critical phase wherein, by chance events, the regime Ri gets a head start

t,ε , j t,ε , j on the alternative: µi > µ −i , j = 1,2 . An implicit assumption is that such a regime may be rooted more firmly in past adaptation than the alternative. Put differently, the former enjoys a wider network of auxiliary institutions that has been built up from past adaptations. In our simple model, those supporting institutions are omitted. But their effects are still being captured by the parameters associated with the lag variables in (10). In particular, if the sum is greater than , then the reinforcing linkages ∑ ρi,k ∑ ρ −i,k k =1, pi k =1, p−i

24

j λ1 of the regime Ri , , j = 1,2 , is greater than that of R−i . Subsequently, the system will (1− ∑ ρi,k ) k =1, pi

j j t,ε , j t,ε , j most likely visit the convention hi with higher frequency than h−i : µi > µ −i , j = 1,2 . This condition allows the self-fulfilling mechanism to take effect to shape the pattern of expectations. The following proposition summaries our discussion3:

Proposition 2: Suppose by adaptation, the regime R1 happens to enjoy sufficiently stronger networking effects than does: . Then, in the long run, the virtuous circle will determine the R2 ∑ ρ1,k > ∑ ρ 2,k k =1, p1 k =1, p2

1 2 1 2 payoff structures so that conventions h1 and h1 become the stochastically stable state of G and G , respectively. Alternatively, if the regime has stronger networking effect: . Then, R2 ∑ ρ1,k < ∑ ρ 2,k k =1, p1 k =1, p2

1 2 1 2 the vicious circle will boost both h2 and h2 to emerge in G and G , respectively.

A key feature of this framework is that individuals have a limited understanding of their environment.

They do not know beforehand which regime will dominate the system. Therefore, they do not know for sure what are the true values of payoffs associated with certain actions. They then must frequently update their assessment about these payoffs and form the expectation about the other players’ action in according to what they have observed. When playing adaptively, they make mistakes, which cause the system to be shaken out of one circle to another. Their adaptive play then creates the context for the two alternative regimes to compete in according to proposition 1. In the long run, when the random shocks taper off to zero, one regime will appear with higher frequency than the other. Hence, individuals will be able to learn about the true value of the payoffs. Thus, the adaptive play in each game will converge to its stochastically

3 For a more detail, see the appendix.

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stable state as if individuals knew in advance what regime would dominate the system. The essence is that, at least in the long run, the conventions of behavior in both games are contingent on the outcome of the competition between the two alternative circles. As time unfolds, the prevailing circle drives the payoff structures and the outcomes in both games. This circle reflects complementarities or networking effects that link different games together. Such a multiple game linkage is not strategic4, but parametric5.

An analogous result for the case of coordination game with non-symmetric payoffs can be presented below.

Provided the payoff structures are no longer symmetric, certain assumptions must be changed accordingly.

j In particular, we still assume that the convention hi , j = 1,2 associated with the regime Ri is

j j characterized by the set of parameters: {λi ,θ i ,(ρi,k ,k = 1, pi )}. Further, the enforcing linkage must work through a specific set of parameters. Consider θ 2 for example, we should have:

∂a 2 ∂θ 2 ∂a 2 ∂θ 2 ii i , −ii −i 2 1 > 0 2 1 < 0 ∂θ i ∂λi ∂θ −i ∂λi

∂b 2 ∂θ 2 ∂b 2 ∂θ 2 ii i , −ii −i (14) 2 1 > 0 2 1 < 0 ∂θ i ∂λi ∂θ −i ∂λi

Proposition 3: By adaptation, if the regime R1 happens to enjoy strong networking effects such that:

j j − j j j − j j j − j a11[θ1 (λ1 )] is sufficiently greater than a22 [θ 2 (λ1 )] or b11[θ1 (λ1 )] is sufficiently greater than

4 This refers to the concept of strategic multi-game linkages by Bernheim and Whinston (1990). 5 I am indebted to Jonathan Levin for this clear description of linkage effects that I try to investigate in this paper.

26

j j − j 1 2 b22 [θ 2 (λ1 )], j = 1,2 . Then the adaptive play will converge to the convention h1 and h1 in the game

1 2 G and G , respectively. Alternatively, if the regime R2 has stronger networking effects, then the vicious

1 2 1 2 circle will boost both h2 and h2 to emerge in G and G , respectively.

The proposition 3 has completed our formal analysis of institutional change in the presence of institutional complementarities. A corollary is that if the inferior regime gets ahead start on an inherently superior one and if by chance events the former enjoys strong networking effects, then as time passes, it gets harder and harder for stochastic forces to displace the inferior regime in favor of the superior one. The system experiences a state of inertia.

VI. Conclusions

In this paper, past adaptations have been embodied in the selection process via reinforcing linkages or networking effects. In the absence of such complementarities, the outcome of the game is defined given the risk factors; and the risk factors are justified given the payoff structure of the game. Under our presumption that players extrapolate across games, payoff structures may not be known from the beginning, but will be revealed by the adaptive process. In this context, a norm or convention can be understood only within a dynamic framework, which explains how expectations are formed and how adaptive play selects a particular set of complementary conventions over the other. Such a regime has emerged because its

27

networking effects brought about by the precedent are more potent than the alternative regime. History therefore matters (North, 1973; Arrow, 1997).

This notion of institutional development combines ideas from the evolutionary theory with the study of institutional adaptation in the presence of complementarities. It suggests that: (1) the stability of a particular norm or institution should not be considered in an idiosyncratic but in synchronic perspective. (2) The better the regime adapts to past conditions, the stronger its networking effects so that this regime still persists even when it is no longer optimal. (3) Institutional change in response to the change in the environments (tastes, technologies, and political conditions), therefore, must take time. Such a change cannot simply be brought about by an ad hoc policy that rapidly erodes the previous expectations, for the need to rely on past adaptations is too important to be ignored. Instead, institutional change may be facilitated by policies that induce the creation of supporting networks to guide expectations and to smooth out temporary difficulties (Arrow, 1997).

Appendix

Sketch of Proof of Lemma 2

Notice that a Markov chain P ε ,2 is irreducible (i.e. the process has unique recurrent class, which is the whole state space X m,2 .) From the fundamental result for a finite Markov chain, it follows immediately that

28

lim µ t,ε ,2 = µ ε ,2 (A1) t→∞

where, µ ε ,2 is the stationary distribution , which describes the time-average asymptotic behavior of the

ε ,2 2 process P independently of the initial state h0 .

Notice also that, P ε ,2 is a regular perturbed Markov process of P 0,2 . Young (1993) shows that there exists

lim µ ε ,2 = µ 2 (A2) ε →0

where, µ 2 is a stationary distribution of P 0,2 . Further, µ 2 (h 2 ) > 0 if and only if h 2 is contained in the recurrent class of P 0,2 , which has minimum stochastic potential.

2 Finally, since we assume that the game has the unique stochastically state, say hi . Then,

2 2 2 2 2 2 µ i = µ (hi ) =1; and µ −i = µ (h−i ) = 0 .

We should note that Kandori, Mailath, and Rob (1993) obtain a similar result for symmetric coordination games by using somewhat different dynamic process. In their model, there is a single homogeneous

population of N agents who play a symmetric 2 × 2 game. At each period t, let zt be the current number

of players who play strategy 1. Let π i (zt ),i =1,2 be the expected payoff to strategy i in state zt . The

dynamic process is defined by zt+1 = f (zt ) such that

29

for all 0 < zt < N, zt+1 > zt if and only if π (zt+1 ) > π (zt )

They also assume that, with probability ε , each player may switch from playing strategy 1 to strategy 2 or vice versa.

In such a two-state Markov chain, let p denote the probability of transition from convention h1 to

[ N /(1−α )] [ N / α ] convention h2 ; and p’ from h2 to h1 . The order of p and p’ are ε and ε respectively, so

p / p'→ 0 as ε → 0 , if α >1 − α (or equilibrium (1,1) is risk dominant). Notice that the unique

stationary distribution over h1 and h2 is [ p' /( p + p'); p /( p + p')]. It follows that, as ε → 0 , the stationary distribution approaches [1,0]. In other words, the process puts all of the probability on the risk

2 ε ,2 dominant equilibria as ε goes to zero. Apply this result for the game G , we shall see that lim µi =1if ε →0 equilibrium (i,i) is risk dominant.

Proof of Proposition 2

It is assumed that in the beginning of the adaptive process, in each game, there are enough random shocks to shake the process out of one convention to another from time to time. It then appears the situation that

j j − j − j while the convention hi dominating the game G , the convention h−i dominates the game G .

According to Proposition 1, if there is any chance for a more stable pattern of expectations to be taken

1 2 shape, one regime, say Ri = {hi ,hi }, has to overcome the alternative regime, R−i . In other words, the

t,ε , j t,ε , j j system visits the regime Ri with higher frequency than the alternative: µi > µ −i in G , j = 1,2 .

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Next, from the learning rule (12), players’ assessments about the payoff structures in both games are updated accordingly. Specifically, in both following equations, the first term gains more weight than the second one:

− j t,ε , j − j − j j t,ε , j − j − j j ai (ε,t) = µi ai [θ i,t (λi )] + µ −i ai [θ i,t (λ−i )], j = 1,2

− j t,ε , j − j − j j t,ε , j − j − j j a−i (ε,t) = µi a−i [θ −i,t (λi )] + µ −i a−i [θ −i,t (λ−i )], j = 1,2 (A3)

∂a − j ∂θ − j ∂a − j ∂θ − j Given i i and i i , we then see that the value of − j tends to increase − j j > 0 − j j < 0 ai (ε,t) ∂θ i ∂λi ∂θ −i ∂λ−i

− j − j and the value of a−i (ε,t) tends to decrease, or equivalently, the risk factor of hi tends to become lower

− j 6 and that of h−i tends to become higher . Subsequently, the learning process is self-fulfilling in the sense

j that, when random shocks become smaller, it makes harder for the system to visit the convention h−i in

G j , j = 1,2 . Thus,

t,ε , j t,ε , j µi → 1;µ −i → 0, j = 1,2 , when t → ∞;ε → 0 .

By assumption, we also have:

λ j θ − j (λ j ) → θ − j (λ j ) =θ − j + i , j = 1,2 , when t → ∞ . m,t i m i m,t0 (1− ∑ ρi,k ) k =1, pi

6 See (6) with notice that players now are heterogeneous with respect to payoffs. For a more detail, see Young (1998, Ch. 5).

31

Subsequently,

− j − j − j j ak (ε,t) → ak [θ k (λi )], j = 1,2,k = 1,2, when t → ∞;ε → 0 (A4)

We have not address yet the question why one regime, Ri , may appear with higher frequency than the alternative.

j j j − j j According to (11), ai tends to converge to, while fluctuating around, its mean ai [θ i (λi )] and a−i tends to converge to, while fluctuating around, j j − j . Suppose the sum is a−i [θ −i (λi )], j = 1,2 ∑ ρi,k k =1, pi greater than , or equivalently, the reinforcing linkages of the regime , ∑ ρ −i,k Ri k =1, p−i

j λ1 j j − j , j = 1,2 , is greater than that of R−i . Then by construction, ai [θ i (λi )] is greater than (1− ∑ ρi,k ) k =1, pi

j j − j j a−i [θ −i (λi )], j = 1,2 . If the slack is sufficiently large, the equilibrium hi becomes risk dominant in the

j 7 j j game G , j = 1,2 . This condition implies that the system will visit hi , j = 1,2 more often than h−i

t,ε , j t,ε , j when random shocks become smaller: µi > µ −i . It in turn allows a stable pattern of expectations to

be taken shape, as we mentioned above. Subsequently, the regime Ri will most likely emerge in the long run in a self-fulfilling manner.

Our analysis does not rule out the possibility that, by chance events, the regime R−i might still get start

t,ε , j t,ε , j ahead on Ri : µi < µ −i and by the updating rule (12), that regime R−i might still emerge in self-

fulfilling pattern to dominate the system, despite its reinforcing linkages are weaker that those of Ri . We

7 See Heterogeneity in payoffs in Young (1998, Ch. 5).

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also do not rule out the possibility that the system might never get to a stage to form a stable pattern of expectations. But our analysis emphasizes that the larger the slack: > , the more likely ∑ ρi,k ∑ ρ −i,k k =1, pi k =1, p−i

it is the regime Ri will overcome the alternative regime. In the long run, it determines the asymptotic outcomes in both games.

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Table 1: Retirement Income

The Young Farmer:

The Old: New Tech Old Tech

Use Saving f (kn ) −π (kn ) , π (kn ) β , π (1,ko )

, , Use Expertise π (2,kn ) −π (1,kn ) π (1,kn ) −α π (2,ko ) −π (1,ko ) π (1,ko )

Table 2: Community Risk Sharing

Player X (Young):

Player Y (Old): Invest Don’t Invest

µ − p , µ − p −τ (σ 2 ) 0, 0 Require Transfer y x xa

0, 0 µ − p , π (1,k ) Don’t Require y o

36