Learning dynamics, genetic algorithms, and corporate takeovers

Journal of Economic Dynamics & Control 24 (2000) 189}217

Learning dynamics, genetic algorithms, and corporate takeovers Thomas H. Noe *, Lynn Pi
A.B. Freeman School of Business, Tulane University, New Orleans, LA 70118-5669, USA
Department of Management & Finance, California State University, Hayward, Hayward, CA 94542, USA

Abstract This paper simulates, via a genetic-learning algorithm, free-riding and coordination failure when shareholders are confronted with an unconditional tender-o!er bid between the pre-takeover and post-takeover value of their "rm. The outcomes produced by the simulations o!er strong support for the hypothesis that coordination to tendering strategies permitting o!er success is impaired by increasing the number of shareholders and the divisibility of shareholdings. Further, the outcomes of the simulations closely conform to the restrictions imposed by the Nash equilibrium hypothesis. When the number of shareholders and the disability of shareholdings are both small, the aggregate outcomes of the simulations converge to the aggregate outcomes produced by e$cient Nash equilibria. Otherwise, the outcomes of the simulation more closely resemble the outcomes of ine$cient Nash equilibria.  2000 Elsevier Science B.V. All rights reserved. JEL classixcation: C7; C15; G34 Keywords: Takeovers; Free-rider; Tendering strategies; Genetic algorithm

1. Introduction The question of determining the rational response of a shareholder to an unconditional tender o!er priced between the pre-takeover and post-takeover

* Corresponding author. Tel.: #1-504-865-5425; fax: #1-504-865-5425. E-mail address: [email protected] (T.H. Noe) 0165-1889/00/$ - see front matter  2000 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 5 - 1 8 8 9 ( 9 9 ) 0 0 0 0 4 - 4

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value of the "rm is one of the most important unresolved questions in corporate "nance theory. O!ers in this range are clearly feasible. Thus, any theoretical model of takeovers should allow raiders to make o!ers in this range. Even if all one wants to accomplish is to show that making o!ers in this range is suboptimal, the modeler must "rst determine the payo!s that will result from such o!ers in order to compare them with those induced by o!ers from other ranges. As well as being very important, this question is a di$cult one to answer. If one assumes that shareholders act in a non-strategic fashion, then no Nash equilibrium vector of shareholder responses exists. The logic underlying this result is simple. First, note that no equilibrium exists in which the o!er fails. For, if shareholders believe that the o!er will fail regardless of their actions, then it is in their individual interests to tender into the o!er, since, by tendering, they receive the tender o!er price, which exceeds the pre-takeover value of the "rm. However, if all shareholders tender, the o!er must succeed, contradicting the initial hypothesis of failure. A parallel argument shows that no Nash equilibrium exists in which the o!er succeeds. The non-existence of a Nash equilibrium in the non-strategic setting implies that this paradigm is incapable of providing any insight into the question of shareholder responses to tender o!ers at prices between pre-takeover and post-takeover value. It is possible to skirt this question in models of raider behavior in takeovers by assuming that non-strategic shareholders will reject all tender prices below the post-takeover value of the "rm. Given this conjectured response, the raider will eschew such o!ers, either refraining from making an o!er or o!ering a price at least equal to post-takeover value. These strategies keep the shareholders' responses to o!ers below the post-takeover price o! the equilibrium path and thus support a Nash equilibrium. However, this equilibrium is not subgame perfect. For, if an o!-equilibrium node is reached at which the raider o!ers a price between the pre-takeover and post-takeover values of the "rm, and shareholders conjecture that other shareholders will follow their equilibrium strategy of sco$ng the o!er, each individual shareholder will believe that the o!er will fail, and thus they will "nd it in their interest to tender rather than sco!, contradicting the sequential rationality of the equilibrium strategies. Because subgame perfection appears to be a desideratum for any reasonable explication of the strategic evolution of the tender o!er game, the plausibility of an approach that cannot ensure the existence of subgame perfect equilibria is questionable. While non-strategic models are dogged by the problem of non-existence, strategic models of shareholder tendering yield a surfeit of equilibria. In fact, when the number of shareholders becomes large, equilibria that produce divisions of the synergy gains between the raider and the shareholders approximating any level of raider pro"t between 0 and the entire takeover gain exist (see Noe, 1995). Thus, strategic models utilizing the Nash solution concept also fail to make a determinant prediction regarding raider pro"t in tender o!er contests.

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Given the plethora of Nash equilibria in the takeover game, a natural approach to producing a determinant prediction is to use the literature on equilibrium re"nements to narrow the range of outcomes considered. Re"nements can be usefully divided into those considering the robustness of the equilibria to perturbations (see, e.g., Selten, 1975) and those which consider the incentives of groups of agents to deviate from the equilibrium (see, e.g., Bernheim et al., 1987). Since the takeover game features a plethora of pure strategy equilibria in which all agents are playing strict best replies, it is not possible to re"ne the set of equilibria to a set of reasonable size using perturbation arguments. Coordination arguments, such as coalition proofness point to the e$cient pure-strategy equilibria as the most plausible outcomes of the takeover game. To see this, note that the only way raider pro"t can converge to zero as the number of shareholders increases is for the o!er to fail with a high probability. However, such failure also lowers shareholder pro"ts, on average. Thus, rational coordination among coalitions of shareholders would never lead shareholders to adopt strategies that induce such failures. Therefore, re"nements of the set of Nash equilibria tend to support the robustness of pure strategy e$cient-Nash equilibria featuring high raider pro"ts, even when the number of shareholders is large. In light of the empirical evidence, this result is somewhat disturbing. This body of evidence clearly argues against the ability of raiders to capture a signi"cant fraction of the surplus in corporate takeovers. Further, laboratory experiments with human subjects feature signi"cant probabilities of coordination failure when the number of agents is large (Kale and Noe, 1997). The folk wisdom that coordination problems are severe when ownership is di!use is also quite intuitively plausible to economists, as evidenced by the common recourse made by theorists to this argument when formulating corporate control models (see, for example, Hirshleifer, 1992). Thus, laboratory experiments on human subjects and Nash equilibrium re"nements point to radically di!erent conclusions regarding the plausibility of coordination to e$cient outcomes in actual takeovers featuring a large number of shareholders. What accounts for this divergence? One possible explanation is (a) a discrepancy between actual agent objective functions and pro"t maximization. E$cient pure strategy outcomes sometimes call for identically endowed agents to play di!erent strategies and thus to reap di!erent rewards from takeovers. If human agents are concerned with equitable division of takeover gains, this &extra argument' to their utility function could lead them to eschew &unfair' e$cient outcomes, resulting in less e$cient but &fairer' Nash equilibria.  See Noe (1998) for a formalization of these arguments.  For an example of the use of a fairness argument to explain the outcome of a takeover experiment, see Cadsby and Maynes (1997). For the use of fairness arguments to explain the outcomes of threshold contribution public good game, a game closely related to the takeover game, see Marwell and Ames (1979).

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A second possibility is (b) a perceptual bias on the part of human agents that leads them to ignore the e!ects of their own actions on the takeover outcome. That is, when the number of shareholders is large, individual shareholders fail to perceive their own, perhaps small, ability to a!ect the outcomes of the takeover attempt. This failure will reduce their willingness to tender and thereby will reduce e$ciency. A third possibility is (c) that learning dynamics alone account for this divergence. When multiple optimal strategies for an individual agent exist, then simple, plausible, learning dynamics may not lead the agent to play a single strategy. Instead, adaptation may leave agents with multiple optimal strategies. This variability, while not suboptimal at an individual agent level, has a profound adverse impact on the total pro"ts garnered by raiders and shareholders. Clearly, (a) and (b) are su$cient to explain ine$ciency in takeover o!ers featuring a large number of shareholders. However, are they required to explain the facts? In other words, can (c) alone also explain the failure to attain e$cient outcomes. If hypothesis (c) is su$cient, it has the advantage of economy over (a) and (b). Clear evidence exists that dynamic learning takes place in laboratory experiments. Thus, any theory explaining the evidence must account for learning dynamics. If these same dynamics can explain the long-run deviations from the Nash outcomes, without recourse to invoking the additional hypotheses of (a) and (b), then a more &economical' explanation for subject-tendering strategies can be developed. Empirical observations and laboratory experiments on human subjects cannot determine whether learning dynamics alone can account for tender o!er ine$ciency, because such experiments cannot directly control for, and thus separate out, the e!ects of preference and perceptions, as opposed to the adjustment dynamics, on the behavior of agents. A technique that allows us to "x the preferences of agents and thus isolate the pure e!ect of the learning dynamics is required. In this paper, we attempt to implement such an investigation, using arti"cial agents programmed with a pro"t-maximizing payo! function and a &plausible' learning protocol provided by a genetic algorithm. Genetic algorithms, a technique for modeling arti"cial agents developed by Holland (1975), has been shown in many contexts to successfully model human leaning (see Andrews and Prager, 1994). Through a process similar to natural

 This argument is the classic atomistic actor argument of Grossman and Hart (1980).  Our e!ort to explain the divergence of share-tendering behavior from Nash outcomes via learning dynamics resembles Samuelson's (1997) e!orts to show that the divergence between the outcomes of the Ultimatum Game and the subgame-perfect Nash equilibrium outcome can be explained using evolutionary arguments without making recourse to subject's sense of &fairness'.

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selection in population genetics, the genetic algorithm simulates learning and adaptation. The outcomes produced in the simulations support the hypothesis that learning dynamics alone can explain the reduced e$ciency of tender o!ers when the number of shareholders is large. In no case, do treatments featuring a large number of agents converge to pure-strategy, e$cient Nash equilibria. Instead, they feature a signi"cant randomization by agents and non-trivial probabilities of takeover failure. In contrast, when the number of shareholders is small, learning usually leads to the attainment of pure-strategy e$cient outcomes, even when such outcomes call for the arti"cial agents to play highly asymmetric strategies. The failure of the simulation to attain the e$ciency produced by pure-strategy Nash equilibria does appear to represent a failure of the simulation results to conform to the general restrictions imposed by the Nash equilibrium conditions. In fact, the fraction of shares tendered conforms closely to the fraction implied by the Nash equilibrium hypothesis. Further, in simulations featuring a large number of shareholders (more than 50), the relationship between raider and shareholder pro"ts almost perfectly conforms to the relationship imposed by the Nash equilibrium conditions when ownership is di!use. This paper is organized as follows. Section 2 of the paper presents the basic model of shareholder-tendering behavior. Section 3 describes the genetic algorithm model of strategic evolution in detail. Section 4 presents the results of the genetic algorithm simulations and compares the raider and shareholder payo!s with those induced by the entire Nash equilibrium set of the game, as well as with some of the speci"c equilibria discussed in the takeover literature.

2. The model 2.1. Basic dexnitions Consider a "rm with n shareholders; let N"+1, 2, 3,2, n, represent the set of shareholders. Each shareholder, i3N, holds h shares. Let H "+0, 1, 2, 3,2, h ,; G G G let H"“L H . Let h represent the vector (h , h ,2, h ), i.e., h represents the GG G   L number of shares held by each shareholder. Thus, we will call h the vector of shareholdings. Each shareholder decides the number of shares he will tender, c . G A pure strategy for agent i, c , is an element of H . A strategy vector for the game G G  This paper is not the "rst to use genetic algorithms to simulate strategic behavior. Arifovic (1994) uses genetic algorithms to simulate strategic interaction and dynamic learning in a cobweb model of price determination. Nor is our paper the "rst to apply genetic algorithms to a problem in "nancial economics. Allen and Karjalainen (1994) use genetic algorithms to identify technical trading rules.

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is a map c : NPH. For c3H and s3H , let (s , c\G)3H represent the strategy G G vector obtained from c by deleting the strategy of agent i and replacing it with s , G i.e., (s , c\G)"(c , c ,2, c , s , c ,2, c ). Let N represent the natural numG   G\ G G> L bers, and let N "N6+0,.  The shareholders must decide how to respond to an unconditional tender o!er from a raider. The o!er will succeed if ¹ or more shares of the "rm are tendered. Thus, the fraction of the "rm's shares that the raider must purchase to obtain control is given by a,¹/ L h . All tendered shares are purchased by G G the raider at a bid price of b. If the o!er succeeds, the share price increases from v , the value of the shares under the incumbent management, to v , the value of   the "rm if managed by the raider. To keep the notation simple, we normalize the gain from the takeover by assuming that the value of the "rm under incumbent management is 0 and that the value under the raider is 1, and that b3(0, 1). Given this normalization, the bid price in our model should be interpreted as the fraction of the total takeover gain, v !v , impounded in the bid price. The   payo! to shareholder i, if he selects pure strategy s 3H , and other shareholders G G follow c3H is given by



if s # c (¹, G H H$G bs #(h !s ) if s # c 5¹. G G G G H H$G The "rst branch in the de"nition of the shareholder's payo! above represents the shareholder's gain if the o!er fails } the sum of the payments for tendered shares, bs , and the value of the shareholder's remaining shares, given that incumbent G management retains control, v (h !s )"0(h !s )"0. Similarly, the second  G G G G branch represents the shareholder's payo! if the o!er succeeds } the sum of the payments for tendered shares bs and the value of the shareholder's remaining G shares, given that the raider takes control, v (h !s )"1(h !s )"h !s . We  G G G G G G assume that when all other shareholders tender, one shareholder alone cannot block or insure the takeover's success. Formally, we assume that for all i3N, h 4¹ and h 5¹. G \G H u (s , c\G)" G G

bs G

2.2. Nash equilibria To de"ne shareholder payo!s when randomized strategies are employed, we require more de"nitions. Let D(H ) represent the set of probability measures G over H . A mixed strategy for agent i, denoted by k , is an element of D(H ). Let G G G D(H) denote the space of product probability measures de"ned on H, i.e., D(H)"D(H )D(H )2D(H ). For all i3N and strategy vectors l 3D(H ),   L G G de"ne supp(l )"+c 3H : l (+c ,)'0,. Let ; (k) represent the utility of agent G G G G G G i under the strateg y vector k3D(H ), i.e., ; (k)"u (c) dk(c). For each i3N,  G G

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we let D(H\G), D(H ) represent the marginal distribution determined by H$G H the tendering strategies of all shareholders except i. For each strategy vector s 3H and strategy vector k3D(H), let (s "k\G) represent the mixed strategy G G G vector obtained by replacing the ith component of k with a probability measure concentrated on s , i.e., (s "k\G )"(k , k ,2, k , d G, k ,2, k ), where d G is the G G   G\ Q G> L Q Dirac measure placing all its weight on the pure strategy s . Thus, the payo! to G shareholder i, given that other shareholders follow k, and shareholder i uses pure strategy s , is given by ; (s "k\G), where G G G



; (s "k\G)" u (s , c\G) dk(c)"bs #(h !s )FM \G(¹!s ), G G G G G I G G G and FM \G: NP[0, 1] is de"ned by I





FM \G(m),k c3H: c 5m . H I H$G Here F\G(m) is just the probability, under k, that the sum of the shares tendered I by all shareholders other than i equals or exceeds m. A pure strategy, s 3H , is G G a best reply to k3D(H) if ; (s "k\G)"Max ; (r"k\G). G G G PZ&G A mixed strategy, l 3D(H ), is a best reply for i against k3D(H) if and only if, for G G all s 3supp(l ), s is a best reply to k. In this case, we write l 3BR (k). The basic G G G G G objective of this paper is to determine which strategy vectors k3D(H) represent candidates for rational actions in the takeover game. To determine the static equilibrium points for this game, we will utilize the Nash equilibrium concept, de"ned formally as follows: a mixed strategy vector kH is a Nash equilibrium if, for all i3N, kH is a best reply to kH. G 2.3. Characteristics of Nash equilibria One of the main objectives of this paper is to compare the results of adaptive learning with the outcomes predicted by the Nash equilibrium solution. To accomplish this objective, we must "rst provide some basic characterizations of the Nash equilibria of strategic takeover games. In subsequent sections, we will examine the extent to which the results of adaptive learning conform to these characterizations. A number of authors have contributed to the characterization of the Nash equilibria of strategic takeover games. The results reported in this literature can be divided broadly into three categories: (i) universal properties of Nash equilibria of strategic takeover games, (ii) limiting properties of such equilibria, and (iii) properties of speci"c types of equilibria.

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2.3.1. Universal properties As stated in the introduction, strategic equilibria of takeover games support a wide range of raider pro"t levels. Nevertheless, all Nash equilibria possess some common features. which are delineated below. Result 1. (Bagnoli and Lipman, 1988; HolmstroK m and Nalebu!, 1992). In all Nash equilibria, (i) the raider's per-share proxt is non-negative and not greater than a(1!b) (the fraction of shares required for control times the fraction of takeover gain not impounded into ower price); (ii) all shareholders earn a payow of at least, b, the fraction of takeover gains impounded in the tender price; and (iii) the probability that the takeover attempt will succeed is never less than b. The intuition behind this result is simply that if the probability of success is less than the fraction of the takeover gains impounded in the tender price, b, the shareholders have an incentive to switch to strategies calling for tendering all their shares, contradicting the Nash equilibrium conditions. The raider's pershare payo! equals the expected per-share capital gain from the takeover (the average of the capital loss when the takeover fails and the gain when the o!er succeeds) times the expected number of shares tendered, plus a &covariance' term, representing the covariance between the fraction of shares tendered and the capital gain or loss on the takeover. The fact that the probability of success is at least equal to the fraction of the gain impounded in the o!er price implies that the expected capital gain is positive. The fact that capital gains are positively associated with large share purchases, implies that the covariance term is positive. Together these facts imply the non-negative raider pro"ts reported in Result 1. 2.3.2. Limiting properties of Nash equilibria The independence (and boundedness) of the randomization strategies utilized by the shareholders permits the use of the Law of Large Numbers to characterize the aggregate properties of the Nash equilibria in the limit, as the number of shareholders increases to in"nity. Result 2. (Noe, 1995). If kH is a Nash equilibrium and the number of shareholders is large, then the fraction of shares tendered will approximately equal the fraction required for control a. Moreover, the following approximate linear equations characterize the relation between per-share raider proxt (n), per-share average shareholder gain (u ), and the probability that the tender ower will succeed (FM H(¹)). I n+a(FM H(¹)!b), I u +b#(1!a)(FM H(¹)!b). I

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Result 2 directly implies the following characterization of the raider pro"t/shareholder pro"t relationship obtaining in the limit in all Nash equilibria. Result 3. If kH is a Nash equilibrium and the number of shareholders is large the following approximate linear equation characterizes the relation between per-share raider proxt (n), and per-share average shareholder gain (u ).




u +b#

1!a n. a

Results 2 and 3 follow because, as the number of shareholders increases to in"nity, the fact that all shareholders must be marginal to some extent in any Nash equilibrium implies that the fraction tendered must converge to the fraction required for control. Because, in the limit, the number of shares tendered is "xed, the equilibrium that obtains a!ects only the probability of success. As both raider and shareholder payo!s depend linearly on this probability in the limit, the approximate linear relations speci"ed in Results 2 and 3 are obtained. 2.3.3. Properties of specixc types of equilibria Strategic takeover games have many Nash equilibria (numbering in the thousands, even when the number of shareholders is around ten). For this reason, researchers have naturally focused their attention on the properties of speci"c subsets of takeover-game equilibria. Typically, the focus is on the economic e$ciency associated with a given subset of equilibria. Purestrategy equilibria are notable in this regard. Bagnoli and Lipman (1988) show that in any pure strategy equilibrium, the takeover succeeds with probability 1, and the number of shares tendered exactly equals the number of shares required for control. The intuition for this result is obvious. If a pure strategy vector speci"es a number of tendered shares in excess of the number required for the raider to obtain control, the number of shares tendered can be reduced without a!ecting the probability that the takeover will succeed. Thus, because posttakeover value exceeds the tender price, the best reply of each tendering shareholder will call for a reduction in the number of shares she tenders, contradicting the Nash equilibrium conditions. Similarly, if the strategy vector calls for fewer shares to be tendered than are required for control, then each shareholder has an incentive to tender more shares, again contradicting the Nash condition. Thus, in any pure strategy Nash equilibrium, the raider's per-share pro"t is exactly equal to the fraction of shares required for control times the di!erence between the tender price and the post-takeover value of the "rm. Given our parametric assumptions, the raider's pro"t is thus given by a(1!b).

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HolmstroK m and Nalebu! (1992) show that if the number of shares held by each shareholder is allowed to grow with the number of shareholders at a su$ciently fast rate, ((n), symmetric equilibria exist in which the raider pro"t and average shareholder pro"t converge to the high level induced by purestrategy Nash equilibria. In the equilibria identi"ed by HolmstroK m and Nalebu!, shareholders minimize the variance in the number of shares tendered, subject to the constraint that symmetric strategies are played. For instance, in a 10 shareholders/5 shares example, where 26 shares are required for control, all shareholders would randomize, using a common probability distribution, between tendering two and three shares. In addition, highly ine$cient mixedstrategy Nash equilibria, featuring a high degree of variance in individual shareholder tendering distributions, exist. In these equilibria, randomization induces frequent o!er failure. In fact, Bagnoli and Lipman (1988) show that there exist mixed strategy equilibria such that, as number of shareholders increases to in"nity, the raider's pro"t converges to zero, and shareholder per-share gain converges to the tender price, b.

3. The genetic algorithm In this paper, we simulate the behavior of investors by playing a &virtual' takeover game on a computer. The actions of the &virtual shareholders' are determined by a set of strategy rules that are updated, using a genetic algorithm. First, more pro"table strategies will displace less pro"table strategies as the dominant source of rules for agents. Then, through random mutation, novel rules are introduced into the mix of potential agent strategies. If this novel rule is adopted and is successful, it may become the dominant rule. The long-run dynamics of such a simulated game can then be compared to the Nash equilibria discussed in Section 2 to shed some light on the strategic analysis of corporate takeovers.

3.1. Framework of the game In the simulated game, each shareholder has h #1 pure strategies. These G strategies represent the number of shares the shareholder tenders. Each strategy can be represented by a binary string of 0's and 1's, generated by the base-2 representation of the number of shares tendered. For example, a strategy of tendering 3 of 7 shares is represented by the string &011'. The possible strategies available to an agent are given by a pool of chromosomes made up of these binary strings of 0's and 1's. Each of these 0}1 bits of the chromosome are called &genes'. The actual tendering decision made by a shareholder is

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determined by randomly choosing one of the chromosomes (a string of genes) from this pool. At the start of each period, t, a new chromosome pool is reproduced from the last generation for each player. The reproduction rate of each chromosome is based on its "tness. The "tness of a chromosome of a given player is simply the payo! to that player if the chromosome had been used in the takeover game in the previous round against the strategies actually used by the other agents. More formally, "tness is de"ned as follows: let A (i, t) represent the decoded J value of the lth chromosome of the ith agent at the start of time t, after the t!1 strategies have been played by all agents but before the chromosome pool is updated. Note that A (i, t)3H . Fitness is de"ned as u (A (i, t)"c\G ), where u ( ) ) is J G G J R\ G the payo! function for shareholder i de"ned in the previous section, and c is R\ the vector of strategies actually used in the previous round. In other words, the "tness of each of player i's chromosome at time t is determined by the payo! player i would have earned had his strategy been determined by the chromosome and all other players had followed their actual t!1 strategies. Reproduction occurs when &clones' of chromosomes from the last pool are made. Each new chromosome can be a clone only of a particular chromosome from the previous generation. The procedure of using relative "tness to select the new chromosome pool for each period is an ordinal-based method known as rank-based selection. This rank-based selection procedure takes into account the "tness ranking of the various chromosomes. The procedure operates as follows: to form the mating pool for the next generation, l-least-"t chromosomes from the old pool are replaced by clones of the l-most-"t chromosomes. For example, if l equals to one, and the old pool consists of four chromosomes, encoded as (001, 010, 100, 101), with relative "tness given by (0.05, 0.25, 0.50, 0.20), then the new mating pool will consist of (010, 100, 100, 101). Mutation is then applied probabilistically to this mating pool to transform the selected strategy rules. Mutation causes the gene to #ip, resulting in a &0' if a &1' originally occupied the position and a &1' if a &0' originally occupied the position. Each gene has an equal probability of being #ipped. The resulting chromosome pools for the players resulting from reproduction and mutation are used for making decisions at time t. This process is iterated for K rounds in each game. The "nal generation yields the reported tendering strategy and gene pool for each agent.  Random choice implies, consistent with the classical GA approach, that good and poor strategies are equally likely to be selected for play. However, as we see below, changes in the underlying pool resulting from removal of poor strategies, will over time, reduce the number of poor strategies in the pool.  Usually, genetic algorithms also incorporate another step in developing the gene pool } crossover. In this step, chromosomes are combined by splicing together their associated binary strings to yield novel strategies (see Holland, 1975). However, in some of our simulations, the feasible strategy

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3.2. Parameterization of the simulation Simulations are conducted for 24 di!erent sets of parameter values corresponding to the number of shareholders and the number of shares owned by shareholders in the takeover game. The number of shareholders varies from 2, 5, 10, 15, 50 to 100, and the number of shares each shareholders owns varies over 1, 3, 7, and 15 shares. In the simulations, all shareholders are identical and hold the same number of shares. To gain control of the "rm, the raider must acquire 50 percent of the total number of shares plus 1 share. Thus, if there are 100 shareholders and each owns 15 shares, the minimum number of shares required for a successful takeover is 751. The only exception is the case of 2 shareholders with 1 share each, where the number of shares required for control is only 50% of the total, i.e., 1 share. This speci"cation implies that pure-strategy Nash equilibria are asymmetric, and thus forces a trade-o! between takeover e$ciency and symmetry. The bid price o!ered by the raider is "xed at 0.5 in all simulations. The chromosome pool of each individual shareholder contains 32 chromosomes. These chromosomes represent 32 possible tendering choices. This number is kept constant throughout all rounds in the simulations. The number of genes varies, depending on the number of shares with which agents are endowed. For example, if each shareholder owns 3 shares, then the number of genes in each chromosome is 2. The chromosome pool of each shareholder in the initial generation (time 0) is preselected so that each agent's chromosome pool features an equal representation of all possible tendering strategies. For example, in the 1 share case, 16 pairs of 0's and 1's chromosomes will be in the pool initially. In other words, each shareholder has an equal probability of tendering 0 or 1 shares. Note that as the number of shares increases, the strategy set is larger, and thus fewer duplicate

(Footnote 7 continued) set for agents is very small (e.g., just two strategies in the 1-share-per-shareholder simulations). In these cases, crossover is meaningless. To make inferences clearer, we wanted to follow the same procedure in all simulations. Thus, we decided to drop the crossover step in all of our reported treatments. However, in results unreported we implemented crossover in cases where the strategy set of the agents included at least 5 strategies, and obtained results virtually identical to those reported in the paper.  The bid price was varied in results not reported, yielding virtually identical conclusions regarding coordination and free-riding. However, the split of pro"ts between shareholders and the raider varies with the bid price. Raider pro"ts are higher at lower bid prices. This result is consistent with what one would expect in pure strategy equilibria, but inconsistent with expectations if the symmetric mixed strategy equilibria are played. In mixed strategy equilibria, low bid prices engender an increased probability of failure that may more than o!set the gains from lower payments to shareholders. This e!ect engenders non-extremal optimal bid prices.

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chromosomes are in the pool. For example, when each agent owns 15 shares, 16 di!erent choices are available, ranging from tendering 15 shares to not tendering any shares. So, only two identical sets of chromosomes are in the pool, and the shareholder has a 1/16 initial probability of choosing any one speci"c tendering strategy. The next generation of each player's chromosome pool is made up of copies of chromosomes in the last period, using the selection algorithm. The new pool is formed by partial replacement with elitism. The "ve chromosomes that yield the lowest pro"ts are dropped and are replaced by clones of the "ve with the highest pro"ts. After the mating pool is formed, mutation is applied to the chromosomes probabilistically. For each chromosome there is a probability that its gene(s) will be #ipped. That is, if the chosen gene is originally 0, it will become 1, and vice versa. Since probabilistic mutation is applied to each gene of the chromosome, mutation probability is varied with the length of chromosomes to keep the overall expected number of mutations in each chromosome constant and equal to approximately 0.0031. For example, in the case of 1 share, since only 1 gene is in each chromosome, the probability that the gene will be #ipped is 0.0031. When two genes are in each chromosome, i.e. in the case of 3 shares, each gene will have a 0.00156 probability being #ipped. The mutation probabilities for each gene are then 0.00104 and 0.00078 respectively for chromosomes containing 3 genes and 4 genes. The mutation rate is halved after 50 rounds to reduce the e!ect of mutation noise on the reported results. After all these genetic operations have been applied to the chromosome pool of each agent, a strategy is picked randomly from each agent's pool. This strategy represents the agent's response to the takeover o!er in time t. This process is iterated for 100 rounds (100 times) to yield a "nal tendering strategy for each agent. Results reported in the tables are the outcomes after 300 runs of the stimulations. The exceptions occur when the number of agents is small (2 or 5 shareholders). Then, only 100 simulations are conducted to yield the "nal results.

4. The results 4.1. Convergence To assess the validity of our simulations, we produce statistics and graphs that report convergence of the algorithm. To conserve space, we report these

 In results not reported, we repeat the simulations with random initial chromosome pools and obtain virtually identical results.  We thank a referee for suggesting lowering the mutation rate in later rounds.

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Table 1 Convergence of the chromosome pool Change in tendering distribution across rounds Round 60}70 Round 70}80 2 shareholders/1 share per shareholder 0.0000 0.0000 2 shareholders/15 shares per shareholder 0.0001 0.0001 100 shareholders/1 share per shareholder 0.0127 0.0071 100 shareholders/15 shares per shareholder 0.0073 0.0057

Round 80}90

Round 90}100

0.0000

0.0000

0.0001

0.0001

0.0071

0.0042

0.0046

0.0041

Table 1 reports the average change in the chromosome pool over rounds for selected simulations. The squared di!erence in the proportion of chromosomes representing each of the feasible tendering strategies is used to measure the change in each individual gene pool of each player in each run of the simulation. The largest possible value this di!erence can take on is 1, and the smallest possible value is zero. The resulting numbers are then averaged across players and runs to compute the average change.

results only for the &extreme' parameterizations used in the simulations: the 2 shareholders/1 share per shareholder, 2 shareholders/15 shares per shareholder, 100 shareholders/1 share per shareholder, and 100 shareholders/15 shares per shareholder cases. These results are reported in Table 1. Table 1 reports the average change in the chromosome pool over 10-round intervals from round 60 to round 100 of the simulations. The change in the individual gene pool of each player in each run of the simulation is measured by the squared di!erence in the proportion of chromosomes representing each of the feasible tendering strategies. For example, consider the 2 shareholder/1 share per shareholder case. Consider the "rst shareholder in the "rst run of this simulation. Suppose that at the start of round 60, 24 of the shareholder's 32 chromosomes called for tendering 0 shares and his 8 remaining chromosomes called for tendering 1 share. Suppose also that at the start of round 70, 16 chromosomes called for tendering 0 shares and 16 chromosomes called for tendering 1 share. Then, the change in the tendering distribution for player 1 in run 1 from rounds 60 to 70 would be (24/32!16/32)#(8/32!16/32)"0.125. Note that the largest possible value this di!erence can take on is 1, and the smallest possible value is 0. The changes, computed as outlined above, are averaged across players and runs to compute the average changes reported in the table. As one can see by inspecting the table, the changes are quite small in absolute magnitude (the largest is 0.0042) and are generally decreasing as the

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Fig. 1. Results of 10 randomly selected simulations in the 100 shareholders/15 shares-per-shareholder treatment. The hoizontal axes represents the rounds of the simulation and the vertical axes represents the fraction of shares tendered. The "gure shows that, for the selected sample of simulations, round-to-round variability in the outcome was less in the later rounds than in the earlier rounds.

number of rounds increases. This result provides some con"dence that the convergence properties of the algorithm are satisfactory. In other words, on average, the chromosome pools are converging to a limiting composition. More evidence is provided by the evolution of the per-capita fraction of shares tendered, which is depicted in Fig. 1. As the "gure indicates, this fraction is quite stable over the later rounds of the simulations. Of course, the stability of per-capita fraction of shares tendered does not conclusively show that convergence occurs, even on average, in the individual runs. For example, perhaps, in each run, the fraction of shares tendered changes radically; however, because the changes balance out, the average fraction remains constant. However, the convergence statistic for the chromosome pool reported earlier does show that the chromosome pools of the individual shareholders do tend to stabilize. Thus, radical instability proportion of shares tendered seems unlikely. To provide further evidence on this point, we graph, in Fig. 1, a random sample of runs selected from the 100 shareholders/15 shares treatment. The graph suggests stabilization. It is also worthwhile to note that perfect stability is unrealizable because of the underlying background noise caused by the operation of mutation.

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4.2. Summary statistics A summary of the results of the simulations is presented in Table 2. This table presents per share shareholder pro"t, per share raider pro"t, and the probability of o!er success. Note that the per-share total payo! to all agents from the takeover is equal to the probability of success. Thus, we will use probability of success to measure takeover e$ciency. While e$ciency is measured by the

Table 2 Summary statistics of the takeover results generated in the Genetic Algorithm (a) 2 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5000 0.2500 0.7500

0.0000 0.0000 0.0000

Fraction tendered Raider's gain Shareholder's gain

0.6667 0.3333 0.6667

0.0000 0.0000 0.0000

Probability of success

1.0000

Probability of success

1.0000

7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5714 0.2857 0.7143

0.0000 0.0000 0.0000

Fraction tendered Raider's gain Shareholder's gain

0.5327 0.2617 0.7283

0.0000 0.0025 0.0025

Probability of success

1.0000

Probability of success

0.9900

(b) 5 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.6000 0.3000 0.7000

0.0000 0.0000 0.0000

Fraction tendered Raider's gain Shareholder's gain

0.5333 0.2667 0.7333

0.0000 0.0000 0.0000

Probability of success

1.0000

Probability of success

1.0000

7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5143 0.2571 0.7429

0.0000 0.0000 0.0000

Fraction tendered Raider's gain Shareholder's gain

0.5077 0.2539 0.7461

0.0001 0.0000 0.0000

Probability of success

1.0000

Probability of success

1.0000

T.H. Noe, L. Pi / Journal of Economic Dynamics & Control 24 (2000) 189}217 Table 2 continued (c) 10 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5987 0.2910 0.6923

0.0002 0.0050 0.0033

Fraction tendered Raider's gain Shareholder's gain

0.5333 0.2600 0.7267

0.0000 0.0035 0.0031

Probability of success

0.9833

Probability of success

0.9867

7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5140 0.2489 0.7344

0.0003 0.0041 0.0041

Fraction tendered Raider's gain Shareholder's gain

0.5068 0.2502 0.7431

0.0000 0.0016 0.0017

Probability of success

0.9833

Probability of success

0.9933

(d) 15 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5331 0.2619 0.7281

0.0001 0.0025 0.0025

Fraction tendered Raider's gain Shareholder's gain

0.5111 0.2507 0.7393

0.0000 0.0025 0.0025

Probability of success

0.9900

Probability of success

0.9900

7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5047 0.2474 0.7426

0.0000 0.0025 0.0025

Fraction tendered Raider's gain Shareholder's gain

0.5020 0.2445 0.7421

0.0000 0.0032 0.0034

Probability of success

0.9900

Probability of success

0.9867

(e) 50 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5209 0.2312 0.7088

0.0004 0.0145 0.0139

Fraction tendered Raider's gain Shareholder's gain

0.5067 0.2072 0.6995

0.0001 0.0213 0.0211

Probability of success

0.9400

Probability of success

0.9067

205

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Table 2 continued 7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5028 0.1341 0.6292

0.0000 0.0454 0.0452

Fraction tendered Raider's gain Shareholder's gain

0.5010 0.0971 0.5929

0.0000 0.0535 0.0538

Probability of success

0.7633

Probability of success

0.6900

(f ) 100 shareholders 1 share(s) per shareholder

Mean

Var.

3 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5080 0.1406 0.6161

0.0016 0.0453 0.0476

Fraction tendered Raider's gain Shareholder's gain

0.5019 0.0868 0.5765

0.0003 0.0557 0.0564

Probability of success

0.7567

Probability of success

0.6633

7 share(s) per shareholder

Mean

Var.

15 share(s) per shareholder

Mean

Var.

Fraction tendered Raider's gain Shareholder's gain

0.5000 0.0162 0.5038

0.0003 0.0625 0.0628

Fraction tendered Raider's gain Shareholder's gain

0.5004 0.0053 0.4914

0.0003 0.0629 0.0626

Probability of success

0.5200

Probability of success

0.4967

Note. Table 2 presents a summary of the results of the Genetic Algorithm simulations. Statistics reported are all calculated from the last round of each simulation, averaged over 300 runs of the simulations, in association with the given choice of parameter values. In the cases of 2 shareholders and 5 shareholders, 100 runs of simulations were conducted. Fraction tendered represents the mean fraction of total shareholdings tendered by each shareholder. Similarly, Raider's gain (Shareholder's gain) represents the average pro"t to the raider (average pro"t per shareholder). Probability of success denotes the proportion of successful takeovers in the 300 trials where shares tendered are more than 50%#1 of the total number of shares. The only exception is in the case of 2 shareholders and 1 share per shareholder case in which only 50% is required to make the takeover successful.

probability of o!er success, the other two statistics } per-share raider and per-share shareholder pro"t } measure how the total gain is divided between the raider and the shareholders. Some features are common across all the simulations. Mean per-share raider pro"t is always positive. Although the raider's payo! (expected cash #ow) is positive in all Nash equilibria, outcomes in which raider pro"t is strictly negative are possible (e.g., the raider makes an o!er at a price above the pre-takeover value, but fails to buy enough shares to obtain

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control, and thereby realizes a capital loss on all shares purchased). The highest raider pro"t in any Nash equilibria is given by a(1!b). In no case does the raider pro"t exceed this upper bound. Increasing the number of shareholders generally lowers the probability of success, the raider's pro"t, and the shareholder's gain. These results are consistent with the intuition that increasing the number of agents will tend to increase the likelihood of coordination failure. However, this increase in coordination failure cannot be derived from strategic analysis based on the Nash equilibrium concept. Increasing the number of shares held by each shareholder also adversely impacted coordination and total takeover gains when the number of shareholders was 50 or 100. However, when the number of shareholders was small } 2, 5, 10, or 15 } almost perfect e$ciency was obtained, regardless of the number of shares held by each shareholder. This result is somewhat surprising, especially since, in treatments such as the 2-shareholder/1 share-per-shareholder case, shareholder e$ciency requires one of the shareholders to tender with zero probability and the other to tender with probability 1. Thus, attaining the e$cient outcome requires the gene pool of the two agents to converge to radically di!erent limiting compositions, even through the shareholders' chromosome pools are initially identical. Even in the treatments featuring a large number of shareholders, where e$ciency was low, the summary statistics do not indicate systematic violations of the restrictions placed on aggregate outcomes by the Nash equilibrium conditions. In all Nash equilibria, the probability of success must at least equal the fraction of synergy gains impounded in the tender price (0.50 in the simulations), and in the limit, as the number of shareholders increases to in"nity, the fraction of shares tendered must converge to the fraction sought (50%#1 share in the simulations). An inspection of the numbers in Table 2 shows that these bounds are either respected or else are violated only in the third decimal place. 4.3. Simulation results and asymptotic Nash raider/shareholder proxt equation Table 3 reports the results of OLS regressions of average shareholder pro"t on raider pro"t. As discussed above, although the hypothesis that shareholder strategy vectors are the result of Nash strategies has little predictive power for the level of raider or shareholder pro"t taken separately, it has a great deal of predictive power regarding the relationship between raider pro"t and shareholder pro"t. In fact, as the number of shareholders increases to in"nity, an exact linear relationship between raider and shareholder pro"t holds. Thus, by

 Henceforth, whenever raider and/or shareholder pro"t, gain, or payo! are mentioned in the paper, they are always to be interpreted in the per-share sense.

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Table 3 Comparison of raider pro"t/shareholder pro"t relation produced by the Genetic Algorithm simulations with asymptotic relation characterizing the Nash equilibria of the takeover game (a) 100 shareholders 1 share per shareholder

3 shares per shareholder

Model

b 

b 

b 

Model

b 

b 

b 

1

1.0133 (0.0001) 0.9562 (0.6047) 1.0366 (0.0001) 1.0269 (0.0001)

} } } } !0.8397 (0.0002) !0.4424 (0.0001)

0.4736 (0.0001) } } 0.5249 (0.0742) } }

1

1.0050 (0.0001) 0.9901 (0.3920) 1.0113 (0.0001) 1.0070 (0.0001)

} } } } !0.5759 (0.0120) !0.1757 (0.0001)

0.4892 (0.0001) } } 0.5249 (0.0796) } }

2 3 4

2 3 4

7 shares per shareholder

15 shares per shareholder

Model

b 

b 

b 

Model

b 

b 

b 

1

1.0006 (0.0281) 0.9974 (0.4443) 0.9872 (0.0729) 1.0030 (0.0031)

} } } }

0.4875 (0.0001) } } 0.4196 (0.0001) } }

1

0.9969 (0.8859) 0.9957 (0.7097) 1.0059 (0.0407) 1.0000 (0.3119)

} } } } !0.6387 (0.0052) !0.2232 (0.0001)

0.4861 (0.0001) } } 0.5261 (0.0678) } }

2 3 4

1.0881 (0.0001) !0.1968 (0.0001)

2 3 4

(b) 50 shareholders 1 share per shareholder

3 shares per shareholder

Model

b 

b 

b 

Model

b 

b 

b 

1

0.9677 (0.0001) 0.9171 (0.1561) 0.9917 (0.0001) 0.9779 (0.0001)

} } } } !1.3571 (0.0001) !0.2590 (0.0001)

0.4851 (0.0001) } } 0.5718 (0.0001) } }

1

0.9940 (0.0001) 0.9732 (0.7616) 1.0091 (0.0001) 0.9957 (0.0001)

} } } } !2.2163 (0.0001) !0.1076 (0.0001)

0.4935 (0.0001) } } 0.6327 (0.0001) } }

2 3 4

2 3 4

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209

Table 3 continued 7 shares per shareholder

15 shares per shareholder

Model

b 

b 

b 

Model

b 

b 

b 

1

0.9981 (0.0001) 0.9884 (0.9123) 1.0058 (0.0001) 0.9986 (0.0001)

} } } } !1.6539 (0.0001) !0.0740 (0.0001)

0.4954 (0.0001) } } 0.5990 (0.0001) } }

1

1.0023 (0.0001) 0.9956 (0.5546) 1.0027 (0.0001) 1.0026 (0.0001)

} } } } !0.0875 (0.7081) !0.0699 (0.0001)

0.4956 (0.0001) } } 0.5011 (0.9401) } }

2 3 4

2 3 4

Note. Table 3 represents the results of estimating four models of the relationship between raider pro"t and average shareholder pro"t from the data produced in simulations using the genetic algorithm. In the simulations, all shareholders hold the same number of shares. The number of shares required for control is 50%#1. The share value under the raider is 1, while under the incumbent manager it is 0. The bid price is set at 0.5. The estimation utilizes ordinary least squares regression procedures. The four models estimated are: Model 1: Model 2: Model 3: Model 4:

u "b #b n #e G   G G u "0.50#b n #e G  G G u "b #b n #b n#e G   G  G G u "0.50#b n #b n#e , G  G  G G

where u represents average shareholder pro"t (per share) and n represents the average raider pro"t (per share). In all Nash equilibria of the takeover game, the following relation between raider pro"t and average shareholder pro"t holds in the limit (as the number of shareholders increases to in"nity): u "b#((1!a)/a)n. b represents the bid price and a represents the proportion of shares required for control. Therefore, given the parameter choices used in the simulations, the coe$cient values predicted by the limiting properties of the Nash equilibrium set are b "((1!a)/a), b "0.00 and b "b"0.50.    In the table, the number in parentheses underneath each coe$cient estimate is its p-value under the null hypotheses (two-tailed) described above.

testing the hypothesis that the coe$cients of the estimated regression equation equal the asymptotic-Nash-equilibrium coe$cients, we gain additional insight into the degree to which the outcomes of the simulations resemble those produced by Nash equilibria. Because the linear relationship is predicted by the Nash hypothesis only in the limit as the number of shareholders increases to in"nity, we estimate the regression equations only for treatments in which the number of shareholders is at least equal to 50. A number of regression models are estimated. These models are presented at the end of Table 3. In the "rst regression model, Model 1, both the slope and the

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intercept of the regression equation are estimated. The estimated coe$cients indicate a fairly close "t between the data and the regression line predicted by the Nash hypothesis (see Result 3), although not close enough to prevent a statistically signi"cant variation. Next, by imposing further restrictions on the coe$cients, we test other implications of the Nash equilibrium, hypothesis. The relative excess shareholder gain is the fraction of the gain captured by the shareholders in excess of the tender price, relative to the gain captured by the raider. As shown in Result 3, in any Nash equilibrium the relative excess gain must be approximately equal to a/(1!a), which is approximately equal to 1, given our parameterizations of the simulations. The relative excess shareholder gain in the simulations is estimated in Model 2 by constraining the intercept coe$cient in the Model 1 regression to equal the bid price, 0.50. Under this constraint, b equals the mean relative excess shareholder gain. In all cases, we  could not reject the null hypothesis implied by the asymptotic Nash restrictions at conventional levels of signi"cance. Models 3 and 4 test for non-linearity in the raider/shareholder pro"t relation. The negative-signi"cant coe$cient for b in  the simulations points to a statistically signi"cant degree of concavity in the relationship. Overall, however, if viewed in terms of percentage di!erences, the "t between the asymptotic Nash relationships and the data is good. Moreover, di!erences between the data and these limiting relationships may arise not from the divergence from the Nash equilibrium conditions but rather from the fact that although the number of shareholders is large, it is still "nite. In general, consideration of the above tables points to the tentative conclusion that the simulated tender o!er outcomes roughly conform with the Nash equilibrium restrictions. 4.4. Analysis of shareholder strategies To perform a more comprehensive study, we now directly analyze shareholder strategies. Table 4 presents a summary statistical description of the "nal chromosome pool in each of the simulations. Examining the chromosome pools of the agents allows us to directly examine each agent's mixed-strategy distribution, the statistical distributions used by the agent to generate the tendering strategies she actually plays in the game. Of course, in experiments with human subjects, such direct examination of mixed strategies is not possible. All that one can observe is the realized strategies produced by the underlying mixed-strategy vector. Even assuming that human subjects' mixed-strategy vectors are stationary over time, practical limitations prevent performing enough replications of experiments to utilize the Law of Large Numbers with great con"dence. To understand the results presented in Table 4, note that each chromosome corresponds to a speci"c shareholder-tendering strategy. Thus, the chromosome pool for each agent in each run is described by two statistics: a mean, (k), and a variance, (p). Since one of these statistics can be computed for each agent in

0.5005 0.6661 0.5712 0.5331

k(k)

0.5997 0.5329 0.5141 0.5065

k(k)

0.5199 0.5067 0.5032 0.5018

1 share(s) 3 share(s) 7 share(s) 15 share(s)

(C) 10 shareholders Shareholdings

1 share(s) 3 share(s) 7 share(s) 15 share(s)

(E) 50 shareholders Shareholdings

1 share(s) 3 share(s) 7 share(s) 15 share(s)

0.2443 0.0328 0.0208 0.0178

p(k)

0.2384 0.0417 0.0265 0.0222

p(k)

0.2483 0.0526 0.0230 0.0195

p(k)

0.0053 0.0012 0.0010 0.0008

k(p)

0.0016 0.0004 0.0002 0.0002

k(p)

0.0017 0.0004 0.0002 0.0001

k(p)

0.0004 0.0001 0.0003 0.0010

p(p)

0.0000 0.0000 0.0001 0.0002

p(p)

0.0000 0.0000 0.0000 0.0001

p(p)

0.5335 0.5109 0.5045 0.5021

k(k)

0.6002 0.5331 0.5140 0.5066

k(k)

1 share(s) 3 share(s) 7 share(s) 15 share(s)

0.5115 0.5039 0.5002 0.5004

(F) 100 shareholders Shareholdings k(k)

1 share(s) 3 share(s) 7 share(s) 15 share(s)

(D) 15 shareholders Shareholdings

1 share(s) 3 share(s) 7 share(s) 15 share(s)

(B) 5 shareholders Shareholdings

0.2125 0.0257 0.0159 0.0140

p(k)

0.2473 0.0391 0.0246 0.0217

p(k)

0.2385 0.0488 0.0275 0.0246

p(k)

0.0363 0.0067 0.0062 0.0055

k(p)

0.0016 0.0004 0.0002 0.0002

k(p)

0.0015 0.0003 0.0002 0.0001

k(p)

0.0028 0.0005 0.0013 0.0035

p(p)

0.0000 0.0000 0.0001 0.0002

p(p)

0.0000 0.0000 0.0001 0.0002

p(p)

Table 4 presents summary statistics describing the "nal chromosome pools of the agents in the simulations. Note that each chromosome corresponds to a speci"c shareholder tendering strategy. Thus, the chromosome pool for each player, in each run, is described by two statistics: a mean, (k), and a variance, (p). Since one of these statistics can be computed for each player in the simulation, it is in turn possible to compute the mean, k, and variance, p, of these statistics over the set of players. The column headings in the panel represent the results of these calculations averaged across all the runs of the simulation. For example, p(k) represents the variance, over players, of the mean tendering proportions indicated by the players' last chromosome pools, averaged over all runs of the experiment.

k(k)

(A) 2 shareholders Shareholdings

Table 4 Summary statistics for the "nal chromosome pool in the simulations T.H. Noe, L. Pi / Journal of Economic Dynamics & Control 24 (2000) 189}217 211

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the simulation, it is, in turn, possible to compute the mean, k, and variance, p, of these statistics over the set of agents. The column headings in the panel represent the results of these calculations averaged across all the runs of the simulation. Thus, the four statistics: k(k), p(k), k(p), p(p) represent di!erent characteristics of the agents' strategies used in the game. The "rst statistic, k(k), simply measures the mean fraction of shareholdings across all agents' chromosome pools, averaged over 300 runs of the experiment. The second statistic, p(k), measures the variance of the mean tendering proportions indicated by the agents' chromosome pools, averaged over all agents and all runs of the experiment. Thus, this statistic represents the degrees of asymmetry in the strategy sets across agents. A large value for p(k) indicates that some shareholders end up &specializing' in free-riding, while others are forced to &specialize' in tendering to ensure that the o!er succeeds. The third statistic, k(p), represents the average amount of randomization in shareholder strategies. A large value for k(p) indicates that the algorithm induces nondegenerate mixed strategies. The "nal statistic, p(p), indicates the extent to which the degree of randomization varies across agents. A high value for this statistic indicates that some of the agents are playing pure strategies while others are randomizing in a rather robust fashion. The chromosome pool for a particular treatment is worthy of special note. In the 2 shareholder/1 share-per-shareholder case only two tendering patterns are Nash equilibria: one pure-strategy pattern and one mixed-strategy pattern. Thus, the number of Nash equilibria is su$ciently small to allow us to compare the chromosome pool emerging from the simulation with each of the Nash equilibria. Moreover, the trade-o! between symmetry and randomization is particularly stark in this case because all Nash equilibria must exhibit either a very high degree of randomization or (exclusive) a very high degree of asymmetry. When pure strategies are played, the probability of success equals 1, and the variance of each agent's tendering distribution equals 0. In the mixedstrategy outcome, the probability of success is 0.75, and the variance of each agent's tendering distribution is 0.25. In the simulations, the observed probabilities of success equaled 1. This result is consistent with the pure strategy outcome. The average variance of shareholder-tendering strategies (k(p)) in the simulations was only 0.0017. This result is again quite consistent with the pure-strategy outcome. Since the pure-strategy outcome calls for half of the shareholders to tender, the variance in the expected number of shares tendered (p(k)) equals 0.25 in any pure-strategy equilibrium. The actual value for this statistic, 0.24831, is very close to this value. A number of distinct patterns are apparent in the data as one moves from the 2 shareholder/1 share per shareholder case to cases with higher numbers of shares and/or shareholders. Increasing the number of shareholders increases the observed degree of randomization in shareholder strategies. For example, in the 3-shares-per-shareholder case, the expected variance in tendering strategies increases from 0.0004 with 2 shareholders to 0.0067 with 100 shareholders,

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a 20-fold increase. Although the absolute magnitude of this coe$cient seems small, it is important to note that any non-in"nitesimal variance in tendering strategies drives raider pro"t toward zero in the limit as the number of shareholders increases to in"nity. Thus, these small variations can profoundly impact the aggregate outcome. This profound impact of small variations on raider pro"ts is evidenced by inspection of Table 2, which shows that raider pro"ts fall from 0.3333 with 2 shareholders to 0.0868 with 100 shareholders. Thus, as the number of shareholders increases, selection does not result in as homogeneous a chromosome pool, and this e!ect increases randomization and tilts outcomes toward ine$ciency and low raider pro"ts. Consistent patterns can also be detected when the number of shares held by each shareholder increases. First, the degree of asymmetry in the mean tendering vector, measured by p(k), tends to fall. The drop in p(k) is consistent with the agents' choosing to tender an intermediate fraction of their holdings. By far the biggest drop occurs when shares are increased from 1 to 3. Second, in treatments featuring a large number of shareholders, increasing share divisibility tends to dampen randomization. Of course, this e!ect is absent in treatments featuring a small number of shareholders because little randomization in individual shareholder strategies occurs even when divisibility is absent. These e!ects are further investigated in Fig. 2. Fig. 2 plots the distribution of shares tendered for the &extreme' parameterizations of the simulations: 2 shareholders/1 share-pershareholder, 2 shareholders/15 shares-per-shareholder, 100 shareholders/1 share-per-shareholder, 100 shareholders/15 shares-per-shareholder. These average distribution plots indicate that, in the 1-share-per-shareholder cases, each shareholder plays a pure strategy by either tendering or not tendering. Thus, tendering 0 share or 1 share is about equally represented in shareholders' strategy set. However, in the 15-shares-per-shareholder cases, especially with 100 shareholders, there is a strong tendency toward unimodality in the relative frequency of shares being tendered. The relative frequency peaks between 7 and 8 out of 15 shares being tendered. This result indicates that the strategy of randomizing between tendering all shares or no shares is not common in the chromosome pool, although Nash equilibria exist that support such randomization between extreme strategies. The absence of all-or-nothing outcomes is consistent with HolmstroK m and Nalebu!'s (1992) argument that when shareholdings are divisible, shareholders will concentrate on non-extremal tendering strategies.

5. Conclusion In this paper, we applied a genetic algorithm to the task of simulating the actions of shareholders responding to an unconditional tender o!er priced between the pre-and post-takeover value of the "rm. The aim of this exercise was

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to gain some insight into the classical problem of free-riding and coordination failure in takeovers of di!usely held "rms. The outcomes produced in the simulations o!er quali"ed support for the hypothesis that coordination is impaired by increasing the number of shareholders. In no case do treatments featuring a large number of agents converge to e$cient outcomes. Instead, they feature a signi"cant randomization by agents and non-trivial probabilities of takeover failure. Nevertheless, the results do not support the hypothesis of complete free-riding. Rather, the results support the hypothesis of partially successful coordination. As neither the hypothesis that agent strategies are Nash equilibria nor the hypothesis that strategies are

Fig. 2. (a) 2 shareholders and 1 share case, (b) 2 shareholders and 15 shares case, (c) 100 shareholders and 1 share case, (d) 100 shareholders and 15 shares case.

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Fig. 2. Continued.

the result of atomistic maximizing behavior implies this pattern of outcomes, the genetic algorithm produces results that cannot be explained by either of these hypotheses. However, for those parameters of the share-tendering distribution that do not vary signi"cantly across Nash equilibria (e.g., the proportion of shares tendered), the results of the simulations usually approximate the results predicted by the Nash hypothesis. In the takeover games analyzed in this paper, three multiple Nash equilibria exist. Some of these equilibria feature random agent behavior. Moreover, the outcomes produced in these equilibria are quite sensitive to the degree of randomization in subject strategies. In this context, the analysis showed that the genetic algorithm failed to lead to equilibrium strategy distributions su$ciently

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&pure' to avoid ine$cient outcomes. A natural extension of this analysis would be to simulate games whose outcomes are sensitive to random perturbations of subject strategies but have only pure strategy Nash equilibria. Such studies may also yield results that could explicate the divergence between predictions of theory and empirical reality. For example, consider Rosenthal's centipede game, which has a unique subgame-perfect Nash equilibrium determined by backward induction. The playing of this game by human subjects yields results quite di!erent from the Nash outcome. Simulation of this game via a genetic algorithm could produce, through a variation in the algorithm parameters, insight into the adaptive mechanisms that account for the di!erence between Nash and the experimental outcomes. Other problems, such as accounting for the altruistic bias observed in many public goods experiments and for the somewhat puzzling sensitivity of contribution levels to &irrelevant' aspects of the experimental design, also represent logical targets for explication via genetic simulations.

Acknowledgements We would like to thank Mark Bagnoli, our discussant at the Western Finance Association Meetings, and two anonymous referees for insightful comments on an earlier draft of this paper. We also thank Ricky Ho for helpful research assistance. The usual disclaimer applies. This research was supported, in part, by a faculty development research grant from California State University, Hayward.

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 See Kreps (1992, Chapter 12) for a description of this game and a discussion of the outcomes produced in play by human subjects.

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