Equilibrium Concepts for Boundedly Rational Behavior
in Games1
Markus Pasche, University of Jena, Faculty of Economics
Carl-Zeiss-Str. 3, D-07743 Jena, email: [email protected]
Abstract: The paper extends the Nash equilibrium concept to account for arbitrary behavioral heuristics. Players are allowed not only to choose strategies,
but also to select behavioral rules how to choose strategies. It is argued that
behavioral profiles are in equilibrium if no player can benefit from deviating to
another strategy, another behavioral rule or its parametrization. It turns out that
in general payoff maximization is not a dominant behavioral rule. Furthermore
it is shown that heterogeneous behavior may endogeneously evolve even in case
of a unique symmetric Nash equilibrium.
Keywords: Nash equilibrium, behavioral profiles, heterogenity, bounded rationality.
JEL-Classification: C70, C72
1
Introduction
Starting from Simon´s papers in the mid 1950s a large and growing body of literature aims to develop models of decision making departing from rational choice (cf. e.g.
Rubinstein 1998, Conlisk 1996, Lipman 1995 for an overview). Undoubtedly, the standard theory of rational decision making has generated numerous general results and
valueable insights into decision making, hence it is an indispensable basis of contemporary economic theorizing. However, there are several obstacles and objections which
give support to alternative models of “bounded rationality”: On the line of theoretical
arguments it has to be stated that rational decision making is based on axioms about
preferences which have strong logical implications. It is questionable why these axioms
should be a neccessary ingredient of rationality (Sugden 1985, Sen 1985). Thus the
notion of rationality is extremly narrowly defined.
A further objection is the presupposed ability of agents to deal with infinitely complex decision tasks. Especially in case of strategic interactions even simple decision
problems can create a high complexity since agents have to reason about the reasoning of all other players. In addition there are striking empirical arguments against the
explanatory power of rational choice. From experimental economics literature there
are several stylized facts about deviations from rational decision making like framing
effects, simplifying mechanisms, characteristic biases in probability judgements, and
1
Working Paper Series B, No.2001/03, University of Jena, Faculty of Economics
1
other “anomalies” in decision behavior (cf. e.g. Selten 1998, 1998, Rubinstein 1998,
Conlisk 1996). As long as deviations from the standard model are due to limited cognitive abilities (like memory capacities, complexity restrictions) it is questionable to
attest the agents a deficient rationality (Langlois 1990).
One response to the emprirical objections is the development of different generalizations and extensions of rational decision making, such like subjective expected utility
models, rank-dependend utilities, prospect theory, or local utility functions (overview
in Kischka/Puppe 1992, Fishburn 1994, Schmidt 1998). It turns out that most generalizations empirically do not provide better predictions than the standard theory
(Shoemaker 1991, Harless/Camerer 1994, Hey/Orme 1994). Another approach is to
drop the idea of optimizing agents and to develop alternative hypotheses of decision behavior. This distinction draws back to Simon´s notion of “substantial” versus
“procedural” rationality. However there is no precise borderline between both ways of
research since more and more models of behavioral economics are also based on axioms
and hence describe substantial features of reasoning and decision making. But also
some obstacles of boundedly rational decision making in games have to be considered
(cf. Camerer 1997, Rubinstein 1998): (i) There is a growing plurality of new decision
models, but a lack of general principles and results, (ii) many theories are “armchair”
models with poor empirical evidence and ad-hoc constructions, (iii) most models do
not consider that agents are reasoning about how to decide.
This paper is related to the first and the third point. It is argued that agents also
deliberate how to make decisions which is captured by allowing for choosing different
behavioral rules. It is stated that an explanation why certain behavioral rules are observed and others are not, is related to their different performance. Without theorizing
about inference, learning, or evolutionary selection we develop an equilibrium concept
to give a rationale for behavioral rules. Thus behavioral equilibrium concepts are one
tool to obtain general principles of boundedly rational decision making. The approach
differs from other ones employing bounded rationality into game theory by presuming a
certain decision behavior and then looking for appropriate equilibrium concepts which
account for this type of behavior (e.g. Rosenthal 1989, 1993). Instead, at first a more
general equilibrium approach is developed, and afterwards arbitrary types of boundedly
rational or heuristic behavior can be analyzed whether they are a part of a behavioral
equilibrium profile or not.
2
Behavioral Equilibria Concepts
2.1
Behavioral rules and equilibria
To account for behavioral heuristics departing from payoff maximization we have to
extend the usual game-theoretic setting: Let σi ∈ Si , i = 1, .., n be the strategy of
player i, and (σi , σ−i ) a strategy profile with σ−i as the vector (σj )j6=i and Si as player
2
i´s strategy space. Each player i has a payoff function πi (σi , σ−i ). Furthermore let
S−i = ×j6=i Sj be the strategy space of all other players. We confine ourselves to the
case of complete and perfect information. A rational agent will maximize his payoffs
by choosing σi∗ ∈ argmaxσi ∈Si πi (σi , σ−i ), i.e. he will always choose best response, σi∗ ∈
fim (σ−i ) = BRi (σ−i ) with fim : S−i → BRi as a correspondence which maps the
observed strategies of the other players to the set of best responses BRi = {σi ∈
Si |πi (σi , σ−i ) ≥ πi (σi0 , σ−i ) ∀σi0 ∈ Si }.
Other goals of intentional decision making (like satisficing or achieving a certain market
share,) are also affected by the choice of other players. Hence, each behavioral rule has to
take the strategic interdependency into account. But there is no need to derive the rule
from calculus. Also simple heuristics and automated procedures, habits or norm guided
behavior is possible. An arbitrary decision making procedure or behavioral rule fi is a
correspondence fi : S−i × Φi → Ri with Ri ⊆ Si is the set of all responses according to
the behavioral rule fi . A decision according to this rule is σi∗ ∈ fi (φi , σ−i ) = Ri (φi , σ−i )
where φi ∈ Φi is a vector of behavioral parameters, like an aspiration level, desired
markup or market share. Some behavioral rules may not have such a parameter, e.g.
in case of the minmax rule. Then it is Φi = ∅. The upper index ∗ denotes a σi which
is a response to σ−i in accordance with a behavioral rule, but not neccessarily a best
response. Let Ωi be the set of all available behavioral rules of player i. We restrict
Φi and Ωi to the case {Ωi , Φi } = {(fi , φi )|Ri 6= ∅ ∀s−i ∈ S−i }, i.e. for all behavioral
rules and their parametrizations there always exists at least one strategy σi which is a
response to σ−i accroding to the parametrized behavioral rule. This restriction prevents
players to choose objectives (e.g. aspiration levels) which cannot be acchieved by any
strategy.
A vector (fi , f−i ) with f−i = (fj )j6=i and fi ∈ Ωi ∀i is called a behavioral profile. In
contrast to the strategy profile which represents players decisions, a behavioral profile
only indicates the type of how decisions are made. Now the basic idea of the Nash
equilibrium can easily be extended to arbitrary behavioral rules. The Nash solution
∗
∗
(σi∗ , σ−i
) is a strategy profile of mutual best responses σi∗ ∈ fim (σ−i
) ∀i. No player
has an incentive to change his (optimal) strategy. For arbitrary behavioral rules the
equilibrium concept is analogously defined.
∗
) is called a behavioral equilibrium if σi∗ ∈
Definition 2.1. A strategy profile (σi∗ , σ−i
∗
fi (φi , σ−i ) ∀i.
∗
A strategy profile (σi∗ , σ−i
) which meets Definition 2.1 is an equilibrium since each
player makes his decision in full accordance with his objectives, given the decisions
of all other players. There is no incentive to change the strategy unilaterally. If all
players choose the payoff maximization rule fim , the parameter vector φi is empty and
Definition 2.1 coincides with the Nash equilibrium. Hence the Nash equilibrium is a
special case of a behavioral equilibrium.
It has to be noted that the behavioral equilibrium is defined for a given behavioral pro3
file (fi , f−i ) with given parametrizations φi , φ−i . Since we made no further assumptions
about the convexity of Si and the continuity of fi neither a behavioral equilibrium nor
a Nash equilibrium as a special case need to exist. But if an equilibrium exists, it can
be perceived by the players only if fi , f−i is Common Knowledge.
Example 2.1. Consider a one-stage 2-player game with a quadratic payoff function
πi = σi − 2σi2 − σi σj , i = 1, 2, i 6= j. Payoff maximization leads to the best response
functions σi∗ = fim (σj ) = (1 − σj )/4 and for the behavioral profile (f1m , f2m ) the unique
Nash solution is σi∗ = 1/5 with the equilibrium payoffs πi∗ = 2/25 i = 1, 2. Instead of
the best response rule fim the players can select a strategy according to a heuristic rule
σi∗ = gi (σj ) = (φi − σj )/2 with φi as a behavioral parameter. Consider the behavioral
profile (g1 , g2 ). With given φ1 , φ2 there is an equilibrium profile σi∗ = (2φi − φj )/3, i =
1, 2, j 6= i. Let φ1 = 2/3, φ2 = 1/2. Then wwe have the behavioral equilibrium (σ1∗ , σ2∗ ) =
(5/18, 2/18) with the equilibrium payoffs (π1∗ , π2∗ ) = (5/54, 1/18). Consider, instead,
the behavioral profile (f1m , g2 ) with φ2 = 1/2. The resulting equilibrium is (σ1∗ , σ2∗ ) =
(3/14, 1/7) with the payoffs (π1∗ , π2∗ ) = (9/98, 1/14).
2.2
Rules with a balanced parametrization
To capture the effects of learning, adaption of the objective levels, behavioral imitation or experimentation, the equilibrium concept has to be extended to permit changes
of the parametrization. In the spirit of the Nash concept we ask for behavioral profiles with parametrizations where no player can achieve higher equilibrium payoffs by
unilaterally choosing another φi . Although there is no need to assume that players
change their goals, it is reasonable to argue that in presence of a competitive market
or an evolutionary selection process players are more or less forced to adapt to better
performing behavioral rules, including their parametrization. Even without such an
external force it is reasonable that players will prefer higher payoffs to lower payoffs
if they are able to learn (or to calculate) them. Consider given behavioral rules with
arbitrary parametrizations φi . Assume that at the first stage of a game the players
select φi (if Φi 6= ∅) and at the second stage the strategies σi are played according to
the parametrized behavioral rule.
Definition 2.2. Assume a given behavioral profile (fi , f−i ). A behavioral equilibrium
∗
∗
) has a balanced parametrization if
(σi∗ , σ−i
), σi∗ = fi (φ∗i , σ−i
∗
∗
φ∗i ∈ argmax πi (fi (φi , σ−i
), σ−i
) ∀i.
φi ∈Φi
Given a profile of arbitrary behavioral rules each player chooses an optimal parametrization, given the optimally parametrized rules of the other players. Only in case of
balanced parametrizations no player can benefit from changing parameters and the
4
strategy. Obviously each φ∗i is a best response to φ−i where the best response map
depends on the chosen behavioral rules.
Example 2.2. Consider the same payoff functions and behavioral rules fim , gi as in
Example 2.1. For the given profile (g1 , g2 ) we have σi∗ = gi (σj ) = (2φi − φj )/3, j 6= i in
equilibrium. Inserting σi∗ (φ1 , φ2 ) into the payoff function πi and maximizing we obtain
the best response functions φ∗i = (2 + φj )/4, j 6= i and the optimal paramertization
φ∗i = 2/3. Hence the balanced parametrized equilibrium is σi∗ = 2/9 with the payoffs
πi∗ = 2/27.
Consider the profile (f1m , g2 ). In the behavioral equilibrium it is σ1∗ = (2 − φ2 )/7 and
σ2∗ = (4φ2 − 1)/7. Inserting these expressions into π2 and maximizing yields φ2 =
5/8. Hence the balanced parametrized equilibrium is (σ1∗ , σ2∗ ) = (11/56, 3/14) with the
corresponding payoffs (π1∗ , π2∗ ) = (121/1568, 9/112).
2.3
Behavioral equilibrium profiles
Now the players are allowed to select the behavioral rules at the first stage of the
game, i.e. they decide how to decide. As we have assumed perfect information, the
players choose a balanced parametrization at the second stage, depending on the selected behavioral profile. Since they anticipate the balanced parametrized behavioral
equilibria, they will choose appropriate decision rules. Lipman (1991) showed that it
is not inconsistent to model boundedly rational non-optimizing decision making by an
optimal choice of the decision rule. Therefore we extend the equilibrium concept to the
selection of rules:
∗
Definition 2.3. A behavioral profile (fi∗ , f−i
) is a balanced behavioral equilibrium
∗
∗
profile, if it constitutes a balanced behavioral equilibrium (σi∗ , σ−i
), σi∗ ∈ fi∗ (φ∗i , σ−i
)
with
∗
∗
), f−i
(φ∗−i , σi∗ ))
fi∗ ∈ argmax πi (fi (φ∗i , σ−i
∀i
∗
∗
), f−i
(φ∗−i , σi∗ ))
φ∗i ∈ argmax πi (fi (φi , σ−i
∀i.
fi ∈Ωi
where for all fi ∈ Ωi
φi ∈Φi
In a balanced behavioral equilibrium profile no player can benefit from unilaterally choosing another (balanced parametrized) decision rule. Of course the selection of
optimal rules and parametrizations does not imply that equilibrium profiles lead to
superior payoffs. Like in case of rational choice, also behavioral rules may yield paretoinferior outcomes. Like the Nash solution also behavioral equilibria and equilibrium
profiles may fail to exist or multiple solutions can arise. The concept of equilibrium
profiles is based on a maximization principle. Hence it is closely related to the idea
of rational agents and should give support to the Nash behavior. In contrast, it turns
5
m
out that best response behavior (fim , f−i
) need not be an equilibrium profile, but there
may be an incentive to select other heuristic rules which outperform fim . Equilibrium profile analysis then serves as a rationale for the usage of simple rules which are
commonly interpreted as a result of bounded rationality. Moreover, one possible result
(see the Example 2.3) is that in a balanced behavioral equilibrium profile the agents
follow different rules even in case of a symmetric game. This may explain empirically
observed heterogenity of behavior. Heterogenity is then the outcome, not a prerequisite
of equilibrium behavior.
Example 2.3. Again, assume the payoff functions and behavioral rules fim , gi from
Example 2.1. For the profile (g1 , g2 ) and (f1m , g2 ) we had already computed the balanced parametrizations and equilibrium payoffs. Allowing players to choose between
{fim , gi } = Ω at the first stage, we obtain a one-shot game with the payoff matrix 1.
The payoffs for the balanced parametrized behavioral equilibria are taken from Example 2.2 (the payoffs are for player 1, for player 2 the payoffs are given by the transformed
matrix for symmetry reasons).
player 2
Matrix 1
player 1
f2m
g2
f1m
2
25
121
1568
g1
9
112
2
27
There are two profiles (g1 , f2m ) and (f1m , g2 ) which meet Definition 2.3. Neither pure
optimization behavior nor pure application of rule gi are balanced behavioral equilibria
profiles. While the heuristic gi does a poor job when it plays against irself, it is able to
exploit the optimization rule which has no free parameter. It is not neccessary that the
gi -player chooses the optimal parameter in order to have an incentive to change from
fim to gi : all parameters φi ∈ (48/80, 52/80) lead to higher payoffs.
Consider an alternative heuristic σi∗ = hi (σj ) = (1 − σj )/φi which is for φi = 4 identical
with the optimization rule fim . Again, the payoffs are parametrized
by φi . Consider
√
∗
the profile (h1 , h2 ). The balanced
parametrization is φi = 2√+ 3,
√
√i = 1, 2 which yields
the equilibrium σi∗ = 1/( 3 + 3) with the payoffs πi∗ = 3/( 3 + 3)2 . Like in the
case of rule gi the payoff is lower than in profile (f1m , f2m ), and playing h2 against f1m
with φ∗2 yields the same payoffs as in case of (f1m , g2 ) (cf. matrix 2). However, there
exist optimal parametrizations that make hi a dominant rule: it is (h1 , h2 ) is a unique
balanced behavioral equilibrium profile in case of Ωi = {fim , hi }.
6
player 2
player 2
Matrix 2
player 1
f2m
h2
f1m
2
25
h1
9
112
121
1568
√
√ 3
( 3+3)2
Matrix 3
player 1
f2m
g2
h2
f1m
2
25
121
1568
121
1568
g1
9
112
2
27
h1
9
112
361
4704
25
336
√
√ 3
( 3+3)2
Matrix 3 contains the payoffs of all possible profiles including the case of playing rule
gi against hj with balanced parametrizations. Allowing for a rule set Ωi = {fim , gi , hi }
leads to three possible behavioral equilibria (g1 , f2m ), (f1m , g2 ), (h1 , h2 ). It turns out that
even in case of symmetric payoffs and an unique Nash equilibrium multiple solutions
with eventually heterogeneous behavior are possible results of the game, if one allows
for alternative behavioral hypotheses.
The rule set Ωi is closed for analytical reasons. Since the payoff functions and the strategy sets are given to the players, the set Ωi is subject to players creativity, cognitive
dispositions or habits, i.e. it is created by the agents. It may change in time by experimentating or learning and is therefore a source of endogenous uncertainty: Player i may
not be sure about player j´s rule set Ωj and what type of rule he is actually playing.
Therefore, the presented equilibria concepts work only in case of well derined and closed behavioral rule sets and perfect information. In case of an open and endogeneous
rules set it may be questioned if (boundedly) rational players will miss the possibility
of anticipating equilibria, since also permanently changing rules, parametrizations, and
strategies may lead to a good performance (cf. Shubik 1996).
Example 2.4. To demonstrate that a small change in the rule set Ωi makes the
equilibria profile obsolete, consider a modified rule gˆi with σi∗ = ĝi (σj ) = (φi − σj )/4
which is equal to optimization in case of φi = 1. Replacing gi with ĝi and computing all
equilibria parametrizations and the corresponding payoffs leads to the payoff matrix
4. Now the behavioral profile (ĝ1 , ĝ2 ) is the unique balanced behavioral equilibrium
profile.
player 2
Matrix 4
player 1
f2m
ĝ2
h2
f1m
2
25
121
1568
121
1568
ĝ1
9
112
28
361
h1
9
112
11767
152100
121
1560
√
√ 3
( 3+3)2
7
3
Related work
Bounded rationality in games is often modelled by incorporating stochastic errors into
the decison making process in different ways. This motivates concepts like tremblinghand-perfectness and related equilibrium approaches (van Damme 1983). Other approaches are related to the complexity of decision rules. Games are played by finite
automata (Abreu/Rubinstein 1988) what reflects the bounded complexity of behavioral rules. Alternatively, the complexity of decision rules can be considered by assigning
them different decision costs (Conlisk 1988). Also different learning procedures and imitating successful players are special types of boundedly rational behavior. Rosenthal
(1993) shows that using rules of thumb leads to a class of equilibria which may be interpreted as equilibria with decision costs. In Rosenthal (1989) boundedly rational players
are assumed to play their strategies with certain probabilities which are proportional
to their performance.
The majority of game theoretic models of bounded rationality start with presuming
special types of decision behavior and then ask for an appropriate characterization of
equilibria. This might be a shortcoming of behavioral game theory since the model
should explain (empirically observed) decision making behavior, e.g. by means of an
equilibrium analysis. It has to be pointed out that the concept discussed in this paper
does neither argue with individual errors in decision making nor with limited complexity
or decision costs. But these items may be added to the approach by specifying the
decision rules.
Behavioral equilibrium profiles imply that agents choose among different decision rules
and that these choices are optimal. The question arises whether it may lead to inconsistencies if player´s optimal decision rule is not to use the maximizing rule f m . Perhaps
also the regress of deciding how to decide may run into logic problems. Instead, Lipman (1991) shows that it is not inconsistent to model boundedly rational agents that
select the optimal decision rule. Furthermore it can be argued that a problem would
only arise in case of a single-stage game where all players are able to compute the
prevailing equilibrium. The notion of a decision “rule”, instead, makes sense if agents
face a certain decision problem frequently. In this case a behavioral equilibrium profile
can be interpreted as an outcome of a learning process or an evolutionary selection
process. This is a common justification also for the Nash solution and related concepts
(the problem of convergence of these processes will not be addressed in this paper, see
van Damme 2000).
This argument challenges the quesetion about the relationship to other evolutionary
equilibrium concepts like ESS and other deterministic evolutionary dynamics (cf. Friedman 1991, Joosten 1996). In the present form the equilibrium Definitions 2.1 and 2.3
differ fundamentally from ESS or related equilibrium concepts. A first reason is, that
equilibrium concepts from evolutionary game theory apply to large groups with random matching, while the present concept is defined for n ≥ 2 players (as a limit case it
8
covers also the decision of a single agent playing against the nature, where σ−i is drawn
from a joint probability distribution). Another point is that evolutionary concepts have
a dynamic background in that the equilibria are fixed points of a dynamic model. The
basic idea of the dynamic process is that the distribution of strategies evolve according
to their fitness what implies an interpersonal comparisn of payoffs. The identification
of payoff and fitness on the one hand, and the aggregation of payoffs of different players
are critical assumptions which are not employed in the present framework. Furthermore
it is often critized that in evolutionary games agents decide completely without deliberating their decisions. The implied learning or imitation process is rarely modelled
explicitely and is hence only a metaphoric interpretation of a biological process. There
are some developments to overcome this shortcoming by creating more general classes of dynamics and to allow for more explicitely defined learning procedures (a short
overview can be found in Joosten 1996). Instead, modelling boundedly rational players
make deliberate intentional decisions, and the equilibrium concept has to take this into
consideration. Thus the behavioral equilibria concepts are closer to a game-theoretic
or economic view of decision making agents.
4
Rationality, Behavioral Rules, and Evolution
Friedman´s (1953) claim, that the agents make decisions “as if” they are utility maximizing decision makers was based on the argument that arbitrary behavioral decision
rules which systematically deviate from maximization will have lower average payoffs.
Therefore they will be ruled out by maximization behavior in a competitive world. The
concept of behavioral equilibria and the simple example given above, however, indicate
that maximization behavior will not outperform other behavioral rules. In contrast,
pure maximization behavior is in general not an equilibrium profile. Similar results are
well known in the literature (Barnerjee/Weibull 1995, Robson 1996, Mainwaring 1997
among others). The behavioral equilibrium framework is a more general approach to
analyse arbitrary behavioral hypotheses in a strategic context, and to explain what
types of rules will emerge under certain conditions. This may serve as an explanation
of several empirically observed rules of thumb. An application of the equilibrium concepts to oligopoly theory which provides a rationale for markup-pricing behavior can
be found in Pasche (2001).
It has to be noted that balanced parametrizations as well as behavioral equilibrium
profiles are based on optimization principles. Does this mean that agents are optimizers “on a higher level” (the level of selecting behavioral rules)? Although higher level
optimization might be a possible interpretation, it is a misleading idea. As long as it
is reasonable to assume the presence of individual learning or inference, a competitive
market or an evolutionary process, it is also reasonable that parametrizations which
yields higher payoffs will rule out other parametrizations, and better performing heuristics will displace less performing ones. Here we follow Friedman´s argument. But it
9
has to be stressed that on the level of rules an optimal choice requires a closed and
well defined set of alternatives rules Ωi ∀i. This is an artificial assumption, made for
analytical reasons only. In real situations, Ωi itself is created by the agent. This point
requires further discussion since the structure of the decision problem itself is part of
agents disposition, in contrast to classical decision theory (cf. MacCrimmon 1999).
It may be questioned whether behavioral equilibrium profiles which rule out optimizing
behavior should be called “boundedly rational”. Or the other way round: Does it makes
sense to identify payoff maximization with “full rationality”, even if the rational player
knows a priori that he may choose better performing heuristics instead? This question is
also addressed in the literature, and it is well known that outcomes of rational behavior
may be inferior. It seems that the standard notion of rationality becomes obscure, and
a broader conception is needed. Protagonists of “bounded rationality” and “behavioral
economics” criticize the extremly narrow interpretation of rationality as maximization
behavior and the logical implications of equilibria concepts for substantial rationality
(Selten 1990). On the other hand, a complete suspension of optimization and equilibria
principles opens a Pandora box and leads to an overwhelming plurality of behavioral
models which lacks common principles and more general insights. The proposed way
to account for different behavioral rules may be a more rigorous way to escape from
this dilemma and to give a “rationale for boundedly rational behavior”.
5
Concluding Remarks
Of course there are several directions how to extend the framework. As a first extension
behavioral adaptation and learning have to be modelled in a dynamic context to study
whether behavioral equilibrium profiles play a role as a fixed point similar to the relation
between the Nash solution and the fixed points of replicator dynamics in evolutionary
game theory.
A further challenge is to incorporate errors which affect the decision making process.
Errors are not only an add-on to the theory but should be an integral part of a positive
theory of decision making (Looems/Sugden 1995). Of course, errors itself are not a valid
“explanation” of decision making, therefore it has to be cared about how errors are
specified. But the characteristic response to errors can have explanatory power (e.g.
Heiner 1988). The performance of behavioral rules is then related to the robustness
against such errors.
For a broader understanding of decision making behavior especially in case of uncertainty, also the problems of information processing and belief formation rules may be
added to the framework (cf. Lipman 1995). In case of uncertainty a belief formation
rule maps the set of information to the set of beliefs: hi : Ii → Bi and the agents choose
their strategies according to their beliefs: fi : S−i × Bi → Si or with g = h ◦ f we have
gi : S−i × Ii → Si . The equilibrium analysis then includes both, decision making and
belief formation rules.
10
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