Learning dynamics,genetic
algorithms,and corporate takeovers
Thomas H. Noe ,Lynn
Pi
1.Introduction
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Grossman and Hart’s (1980) challenged this
“optimistic story”.
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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*
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Assuming that non-strategic shareholders will
reject all tender prices below the posttakeover value of the firm.
Raider will offer a price at least equal to posttakeover value.(No profit)
Price will between pre-takeover and posttakeover
Not subgame perfect.
Non-strategy models are dogged by the
problem of non-existence.
Strategic model
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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
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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.
2.
3.
A discrepancy between actual agent object functions
and profit maximization.(less efficient but fairer)
A perceptual bias on the part of human agents that
leads them to ignore the effects of their own actions
on the takeover outcome.
That learning dynamics alone account for this
divergence.(learning dynamics may not lead the
agent to play a single strategy.)
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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.
Artificial agents programmed with a profitmaximization and a plausible learning
protocol provided by GA.
2.The model
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2.1 Basic Def.
2.2 Nash equilibria
2.3 Characteristics of Nash equilibria
Basic Def.
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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.
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the fraction of the firm’s shares that the
raider must purchase in order to obtain control
is given by P
  T / i 1 hi
n
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All tendered shares are purchased by the
raider at a bid price of b
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).
i
ui (si ,c )  {
bsi
bsi  ( hi  si )
if
if
si   c j  T
j i
si   c j  T
j i
2.3 Characteristics of Nash equilibria
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1.
2.
3.
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.
Universal properties of Nash equilibria of
strategic takeover games
Limiting properties of such equilibria
Properties of specific types of equilibria
Universal properties of Nash equilibria of
strategic takeover games
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(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);
(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
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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)

   F T   b



  b  (1   ) F T   b


Result 3
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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
1 
  b

  
2.3.3 Properties of Specific Types of
Equilibria
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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 pershare gain converges to the tender price, b.
3.The genetic algorithm
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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
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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
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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
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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
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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
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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
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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
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as the number of shareholders increases to
infinity, an exact linear relationship
between raider and shareholder profit
holds.
4.4 Analysis of shareholder strategies
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 ( )
simply measures the mean fraction of
shareholdings across all agents’ chromosome pools
measures the variance of the mean
2
  proportions indicated by the agents’ chromosome
tendering
pools, averaged over all agents and all runs of the
experiment.
represents the average amount of
 2
randomization in shareholder strategies.
indicates the extent to which the
2
2
  of randomization varies across agents.
degree
 
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 
5.Conclusion
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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.