Learning dynamics,genetic algorithms,and corporate takeovers Thomas H. Noe,Lynn Pi

1.Introduction Grossman and Hart’s (1980) challenged this “optimistic story”. The reason is that if a small shareholder knows that her share value will rise due to the improvement in management after the takeover success, she prefers to hold on to her shares rather than selling them. That is, no shareholder sells her own shares when she expects others to sell. This is the free-rider problem in corporate takeovers that Grossman and Hart (1980) discuss. Does the Free-rider Problem Occur in Corporate Takeovers? Evidence from Laboratory Markets*

Assuming that non-strategic shareholders will reject all tender prices below the post- takeover value of the firm. Raider will offer a price at least equal to post- takeover value.(No profit) Price will between pre-takeover and post- takeover Not subgame perfect. Non-strategy models are dogged by the problem of non-existence.

Strategic model A surfeit of equilibria. The synergy gains between the raider and the shareholder approximating any level of raider profit between 0 and entire takeover gain exist Refinement :to narrow the range of outcomes considered

Raider profit can be zero because fail to takeover.However,such failures also lowers shareholders profit. Rational coordination among coalition of shareholder would never lead shareholder to adopt strategies that induce such failures. Refinements of the set of Nash equilibria tend to support the robustness of pure strategy efficient-Nash equilibria featuring high raider profits,even when the number of shareholders is large.

Divergence 1. A discrepancy between actual agent object functions and profit maximization.(less efficient but fairer) 2. A perceptual bias on the part of human agents that leads them to ignore the effects of their own actions on the takeover outcome. (learning dynamics may not lead the agent to play a single strategy. 3. That learning dynamics alone account for this divergence.(learning dynamics may not lead the agent to play a single strategy.)

fix the preferencesisolate the pure effect of the learning dynamics We need a technique that allows us to fix the preferences of agents and thus isolate the pure effect of the learning dynamics is required. profit- maximizationplausible learning protocol provided by GA. Artificial agents programmed with a profit- maximization and a plausible learning protocol provided by GA.

2.The model 2.1 Basic Def. 2.2 Nash equilibria 2.3 Characteristics of Nash equilibria

Basic Def. N = { 1, 2, 3,... n } represent the set of shareholders. Holds hi shares. Let Hi =( 0, 1, 2, 3,.…hi) h represents the number of shares held by each shareholder. Each shareholder decides on the number of shares he will tender, ci.

the fraction of the firm’s shares that the raider must purchase in order to obtain control is given by P b All tendered shares are purchased by the raider at a bid price of b v0 v1 If the offer succeeds, the share price increases from v0, the value of the shares under the incumbent management to v1. Firm under incumbent management is 0 and that the value under the raider is 1, and that b ~(0, 1). (Si, Ci) = (c1, c2,.....ci si, ci+... cn).

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. 1. Universal properties of Nash equilibria of strategic takeover games 2. Limiting properties of such equilibria 3. Properties of specific types of equilibria

Universal properties of Nash equilibria of strategic takeover games a(1 *b) (i) the raider’s per share profit is non-negative and no greater than a(1 *b) (the fraction of shares required for control times the fraction of takeover gain not impounded into offer price); b (ii) all shareholders earn a payoff of at least, b, the fraction of takeover gains impounded in the tender price (iii) the probability that the takeover attempt will succeed is never less than b.

2.3.2 Limiting properties of Nash equilibria If µ¤ 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 α. per share raider profit (π) per share average shareholder gain (u)

Result 3 If µ¤ is a Nash equilibrium and the number of shareholders is large the following approximate linear equation characterizes the relation between per share raider profit and per share average shareholder gain

2.3.3 Properties of Specific Types of Equilibria 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. Bagnoli and Lipman (1988) show that there exist mixed strategy equilibria such that, as number of shareholders increases to infinity, the raider’s profit converges to zero, and shareholder per-share gain converges to the tender price, b.

3.The genetic algorithm a “virtual” takeover game more profitable strategies will displace less profitable strategies through random mutation, there is always a chance that a novel rule will be followed by one of the agents. 3.1 Framework of the game 3.2 parameterization of the simulation

3.1 Framework of the game each shareholder has hi + 1 pure strategies. For example, a strategy of tendering 3 of 7 shares is represented by “011.” The actual tendering decision made by a shareholder is determined by randomly choosing one of the chromosomes (a string of genes) from this pool. The fitness of a chromosome is simply its Payoff rank-based selection making decisions at time t. iterated for K generations in each game.

3.2 Parameterization of the simulation 24 different sets of parameter The number of share-holders varies from 2, 5, 10, 15, 50 to 100 the number of shares each shareholders owns varies between 1, 3, 7, and 15 shares. the raider must acquire 50 percent of the total number of shares plus 1 share. bid price offered by the raider is fixed at 0.5 each individual shareholder consists of 32 strings of genes, which represents 32 possible tendering choices. the number of shares increases, the strategy set is larger,

Since probabilistic mutation is applied to each bit of the strings a strategy is picked randomly from each agent’s pool. This process is iterated for 100 generations 300 runs

4. The results 4.1 Convergence 4.2 Summary statistics 4.3 Simulation results and asymptotic Nash raider/shareholder profit equation 4.4 Analysis of shareholder strategies

4.1 Convergence This provides some confidence that the the convergence properties of the algorithm are satisfactory. the fraction of shares tendered changes radically; however, because the changes balance out, the average fraction remains constant.

4.2 Summary statistics 1 we will use probability of success to measure takeover efficiency. Mean per share raider profit is always positive. The highest raider profit in any Nash equilibria is given by α(1 - b). In no case does the raider profit exceed this upper bound. consistent with the intuition that increasing the number of agents will tend to increase the likelihood of coordination failure. when the number of shareholders was small—2, 5, 10, or 15—almost perfect efficiency was obtained

4.2 Summary statistics 2 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 infinity fraction of shares tendered must converge to the fraction sought (50% + 1 share in the simulations).

4.3 Simulation results and asymptotic Nash raider/shareholder profit equation as the number of shareholders increases to infinity, an exact linear relationship between raider and shareholder profit holds.

4.4 Analysis of shareholder strategies simply measures the mean fraction of shareholdings across all agents’ chromosome pools measures the variance of the mean tendering proportions indicated by the agents’ chromosome pools, averaged over all agents and all runs of the experiment. represents the average amount of randomization in shareholder strategies. indicates the extent to which the degree of randomization varies across agents.

5.Conclusion support for the hypothesis that coordination is impaired by increasing the number of shareholders. the results do not support the hypothesis of complete free-riding. the results support the hypothesis of partially successful coordination.