Dynamic Competition with Irreversible Moves Friedel Bolle and * Yves Breitmoser † European University Viadrina Postfach 1786, 15207 Frankfurt(Oder), Germany email: [email protected]October 12, 2004 Discussion Paper 207 Europa–Universität Viadrina, Frankfurt (Oder) Abstract We define a class of duopolistic “competitive games” (including Bertrand and Cournot competition, public–good games, and rent–seeking contests) and apply it to define a dynamic game where deviations to increasingly competitive strategies are irreversible. There is generally a unique profile of equilibrium payoffs that Pareto dominates the stage game Nash payoffs, and in a wide range of circumstances, this payoff profile is even unique overall. Moreover, we define a generalized repeated game where deviations to increasingly competitive strategies can be made irreversible by the respective player. Under comparably mild assumptions, we find that the sets of equilibria are payoff–equivalent in both kinds of games. Thus, it is irrelevant whether deviations to competitive strategies are irreversible or can be made irreversible. Fi- nally, we define a perturbed infinitely repeated game, where opportunities to restricts one’s strategies occur with arbitrarily small probabilities, and extend the above equiv- alence (without strengthening the assumptions). JEL classification: D40, L10 (Market Structure), D43, L13 (Oligopoly), C73 (Dy- namic Games) Keywords: dynamic oligopoly, tacit collusion, equilibrium refinement, irreversible moves * We would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the support. † corresponding author
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Dynamic Competition with Irreversible Moves · For a wide range of competitive constituent games, the payoff allocations that are equilibrial in IMGs are unique under renegotiation
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∗We would like to thank the Deutsche Forschungsgemeinschaft (DFG) for the support.†corresponding author
1 Introduction
“To burn down bridges passed” is a rather frequently cited metaphor describing an irreversible de-
cision. As a result of this decision, another decision (the previous troops movement) ismadeirre-
versible. This is quite remarkable, since at first glance it seems preferable that decisions be reversible.
However, decision makers can be better off deliberately restricting their options; not only to win bat-
tles, but also to deter entrances of firms into markets (through committing to competitive prices,
capacity levels, or advertisement spendings, e.g. Sylos–Labini, 1962), and to cope with the own
weaknesses of will (Strotz, 1956, Elster, 1984, and many others). Moreover, players can be restricted
unvoluntarily through their emotions (e.g. Elster, 1998, and Loewenstein, 2000) or through persons
that the players have to report to (e.g. share holders). We shall analyze these aspects in a generalized
model of duopolistic competition. Precisely, we analyze which outcomes would result
1. when deviations to more competitive strategies are irreversible,
2. when the players are provided with opportunities to make such deviations irreversible, and
3. when the players are provided with such opportunities arbitrarily rarely.
Below, we define precisely when a strategy would be said to be more competitive than another one,
but basically: the more competitive a strategy the less the opponent’s payoff, the own payoff is
unimodal in the competitiveness of the own strategy, and the best reply functions in terms of com-
petitiveness (i.e. mappings of opponent’s competitiveness to optimal own competitiveness) have a
unique intersection. This intersection is unique even after linearly intrapolating the best–reply func-
tions (note that we assume the strategy sets to be finite) and the unique intersection is in a pure
strategy profile. Simultaneous move games where the strategies can be ranked in this sense of com-
petitiveness are calledcompetitive games.
The resulting set of competitive games includes numerous models that have been studied in in-
dustrial organization (Bertrand/Cournot competition, rent–seeking contests, and public–good games),
behavioral economics, and political economics (e.g. arms races, trade tariffs, and environmental reg-
ulations). Interactions that are not competitive in our sense include coordination games and zero–sum
games. Let us note that competitive games in our sense are not related to competitive games as they
are defined in Friedman (1983) and Kats and Thisse (1992). Their classes of games are to generalize
zero–sum games, instead of generalizing duopolistic models of competition (as our definition).
The implications of deviations being strictly irreversible (point 1) are analyzed inirreversible
move games(IMGs): in each stage the players can repeat the previously played stage strategy or
they can deviate to a more competitive one; the game ends when both players repeat their previous
strategies, and only the finally played strategies are payoff–relevant. These games are related to
monotone games (Gale, 2001; Lockwood and Thomas, 2002), which is discussed below, and to
spectrum auctions (Milgrom, 2000; Bolle and Breitmoser, 2001).
1
For a wide range of competitive constituent games, the payoff allocations that are equilibrial
in IMGs are unique under renegotiation proofness (including all games with strategic complements
Bulow et al., 1985). We assume renegotiation proofness in the sense ofPareto perfectness(Bernheim
et al., 1987): the strategy profiles are required to induce Pareto efficient Nash equilibria in all sub-
games. Thus, we rule out threats of inefficient responses to deviations. Moreover, there is generally
a unique equilibrium where no player is worse off than in the Nash equilibrium of the constituent
game, which might be focal out of the players’ eyes (however, we do not subject this claim to more
powerful concepts of equilibrium selection).
Basically, IMGs describe the dynamics in games where the decisions have sunk–cost character.
These include games of uncovering information, the destruction of (unique) objects, and the commit-
ment of crimes (see also Schelling, 1963). Moreover, IMGs describe the dynamics that would apply
in infinite interactions, even though IMGs themselves are finite (but importantly, the time horizon is
endogenous, i.e. indefinite). For instance, the IMG equilibria are payoff–equivalent to the equilibria
of infinite games where all rounds are payoff relevant when the players do not discount future payoffs
(provided the irreversibilities apply as described above). This is the most significant characteristic
that distinguishes our approach to IMGs from irreversible move games with exogenous and finite
horizons (see Romano and Yildirim, 2004). In the latter model, the players can backward induce that
their opponent would deviate to competitive strategies in the last round, and as a result, collusive out-
comes can not be sustained in equilibrium. Moreover, since there is a last round in finite IMGs, the
respective set of equilibrium payoffs does not generalize to infinite games (similarly to the distinction
of finitely and infinitely repeated Prisoner’s dilemmas). This point is discussed below.
Another infinite game that is closely related to IMGs (as we define them) is the scope of point 2.
We define a generalized infinitely repeated game where the players do not only choose a strategy to
be played for each round (depending on the history of play), but also decide whether the respective
strategies are implemented in short–term contracts or in one–sided long–term contracts. If the latter
option is chosen, the players can deviate from that strategy only to more competitive ones in future
rounds (note that deviations to strategies that the opponent perceives as being more competitive are
supposed to be beneficial to one’s contract partners, as customers or employees).
When we can assume that the equilibrium paths would be “simple” in all subgames (as explained
next), we can show that these generalized repeated games are payoff–equivalent to IMGs (in equilib-
rium). The equilibrium path of a given subgame is understood to be “simple” when there is a unique
stage game strategy profile that is played infinitely often along this path (this profile would be called
“focus point” of the respective subgame). Note that we do not require any kind of consistency of the
focus points over different subgames. Mainly, we want (and need) to rule out circular equilibrium
paths, as those can not be replicated in IMGs; but when they are ruled out, all IMG equilibria have
corresponding equilibria in generalized repeated games, and vice versa.
2
In some circumstances (as infinitely repeated Prisoner’s Dilemmas), the assumption of simple
paths appears rather demanding, since the set of payoff profiles that can be sustained in equilibrium
shrinks dramatically. More generally, though, the stage games would have arbitrarily large strategy
sets (e.g. in Bertrand/Cournot competition, contests, public–good contributions), and then, all payoff
profiles in non–simple paths can be approximated in simple paths. If, furthermore, short–term adap-
tations of the strategies would be associated with costs, then all non–simple paths would be Pareto
dominated by nearby simple paths. In these circumstances, the equilibria in simple paths are even
unique.
We interpret opportunities to sign long–term contract as means to(re)negotiatethe payoff allo-
cation to be sustained along the equilibrium path. Short–term moves, to contrary, allow to defend
long–term allocations against short–term deviations (through temporary retaliations of such devia-
tions). This distinction suggests thatrenegotiation proofnesswould concern only the Pareto effi-
ciency of the long–term commitments in this context. Consequently, short–term retaliations need
not be Pareto efficient. We model that through applying the limit–of–means criterion in the case of
generalized repeated games (i.e. through assuming that the players do not discount future payoffs),
thanks to which temporary retaliations are payoff irrelevant in the long term (but note that even under
the discounting criterion, temporary retaliations are only marginally inefficient).
Given that we apply the limit–of–means criterion, it is even more significant that (for a wide
range of interactions) the payoff–uniqueness of the IMG equilibria continues to hold in general-
ized repeated games (e.g. in all Bertrand competitions, public–good contributions, and sufficiently
symmetric Cournot competitions and contests). For, the combined assumptions of simple paths and
renegotiation proofness merely discretize the sets of payoff profiles that can be sustained in equilib-
rium, but these sets can still be arbitrarily large (they are bounded only by the number of individually
rational and Pareto efficient stage game strategy profiles, which itself is unbounded). Thus, the reduc-
tion of the set of equilibria observed in our model stems from allowing the players to make long–term
commitments, which appears straightforward in a number of circumstances.
Finally, we define a model of repeated games where (for each round) an opportunity to sign a
long–term commitment arises only with some probabilityπ > 0. Here,π is allowed to be arbitrarily
small, and the resulting model is calledperturbed infinitely repeated games. We find that, regardless
of how smallπ is, the allocations that can be sustained in equilibrium are equivalent to those of gener-
alized repeated games, and hence to those of irreversible move games. Thus, marginal perturbations
of infinitely repeated games suffice to obtain the uniqueness of the outcomes that we find for a wide
range of IMGs. This appears interesting, as most other concepts struggle to refine the equilibria of
repeated games.
In Section 2, we shall provide a number of introductory examples (one for each of the most
prominent competitive games) that illustrate our model, our assumptions, and the main aspects of the
equilibrium induction. In this context, we also discuss the various links to previous studies of dy-
3
namic competition and irreversible moves. The two main strings of studies on irreversibilities shall
be introduced already here, however. As mentioned, one of these strings includes studies of finite
irreversible move games with exogenous time horizon Romano and Yildirim (2004) and Yildirim
(2004). We complement these studies by providing the results for several kinds of infinite interac-
tions. Moreover, Feuerstein and Gersbach (2004) study irreversible investment in infinite interactions
and describe equilibria in grim strategies. They show that for discount factorsδ near 1, collusive al-
locations can be sustained in equilibrium. As a complement to that, we showwhichequilibria result
(under renegotiation proofness) in a wider class of games, and how this can be generalized to games
with voluntary irreversibilities. Note also that Feuerstein and Gersbach (2004) survey a number of
additional empirical and theoretical studies on irreversibilities in industrial interactions.
On the other hand, there are studies of irreversible move games (e.g. Lockwood and Thomas,
2002) where precisely the opposite direction of irreversibilities is assumed. Thus, alternative de-
signs of public–good contribution mechanisms are analyzed. The applications of these models (de–
escalation mechanisms) differ from those of our model (strategies to prevent escalation). However,
the equilibrium outcomes in our case generally Pareto dominate theirs, which suggests that it is easier
to prevent escalation than to de–escalate conflicts. This is discussed in detail below (Section 2).
In Section 3, the competitive games and based on them irreversible move games are defined, and
their equilibrium paths are analyzed. In Section 4, we show that irreversible move games are (payoff–
) equivalent to generalized repeated games and perturbed repeated games. Section 5 concludes, and
some of the proofs are relegated to the appendix.
2 Some Introductory Examples
In this section, we give a number of examples that introduce our approach to irreversibilities and
infinitely repeated games intuitively, and that allow us to differentiate our model from previously
published ones. For most of this section, we focus on irreversible move games (IMGs), whereas
the relation of IMGs to generalized or perturbed repeated games is described below. To follow the
arguments in this section, the definitions of “competitiveness” and IMGs as they were provided above
(in the introduction) are sufficiently precise, but note that formal definitions are provided in Section
3.
Further terms that we use (and that are defined formally in Section 3) arestage strategyand
endpoint. We formalize the definition of IMGs by defining a tree of stages, and each stage is defined
through the history of previous strategy adaptations. In each stage the players move simultaneously,
and the sets ofstage strategiesare the strategies of the constituent game that can be played in a
given stage. The path of play is the sequence of stage strategies that is implied under a given IMG
strategy profile, and itsendpointis the profile of stage strategies that is played finally. In Section
4
4, we introducefocus pointsas the corresponding profiles in repeated games: the focus point is the
(assumingly unique) strategy profile that is played infinitely often along the equilibrium path of a
given subgame. Under the limit–of–means criterion, the focus points are uniquely payoff–relevant
in given subgames of repeated games. Finally, let us clarify that when we refer to renegotiation
proofness, we understand it in the sense of Pareto perfectness (Bernheim et al., 1987): the strategy
profiles are to induce Pareto efficient Nash equilibria in all subgames.
2.1 Repeated Prisoner’s Dilemma
In Prisoner’s Dilemmas (PDs) there are only two choices, cooperation (c) and defection (d), and we
defined to be more competitive thanc. In the following, the payoff relations are understood in one
of the usual ways, e.g. 3,2,1,0. In IMGs, the unique Pareto perfect equilibrium implies(c,c). For,
any deviation from(c,c) would result in the endpoint(d,d), and thus, a coordinated deviation from
(c,c) is Pareto dominated, and unilateral deviations imply losses to the deviating players.
In an infinitely repeated game with arbitrarily small time preferences (or under the limit–of–
means criterion), the Folk theorem of Fudenberg and Maskin (1986) applies. Thus, for each of the
payoff allocations indicated in Figure 1 there is a subgame–perfect equilibrium (SPE) where the
respective allocation describes the mean payoffs along the equilibrium path (notably, in this figure
we allow for circular paths). There are four distinguished points in Figure 1.A would result from the
permanent play of(c,d), B from (c,c), C from (d,c), andD from (d,d).
When we restrict the attention to simple paths, only the payoffs associated withB andD can
be sustained in equilibrium. When we assume Pareto perfectness and allow for circular paths (of
PDs), we find that all outcomes on the Pareto frontier can be sustained (Farrell and Maskin, 1989;
technically, these equilibria are shown to be strongly perfect in the sense of Rubinstein, 1980, and
hence they are Pareto perfect). When we assume simple paths and Pareto perfectness, the unique
corresponding outcome implies the permanent play of(c,c).
Thus, it appears that the assumption of simple paths is already sufficient to obtain the equivalence
Figure 1: Possible Equilibrium Payoffs (in Circular Paths) in Infinitely Repeated PDs
- profit 1
6
profit 2
rD
rA
rC
rBXXXXCCCC XXXX
CCCC
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of repeated games and IMGs. This is true in the case of PDs, but it is not generally. When there are
several Pareto efficient and individually rational stage game strategy profiles, then there are also
several renegotiation–proof equilibria in simple paths—but only one them corresponds with an IMG
equilibrium. Here, it is relevant whether the players are provided with opportunities to restrict their
strategy sets (however unlikely these opportunities are). In these circumstances, the perturbation of
repeated games provides highly significant refinements upon renegotiation proofness (and even upon
renegotiation proofness in combination with simple paths).
Secondly, we see that the assumption of simple paths is necessary. For, players who can restrict
themselves in repeated games would not restrict themselves to playd in the above (circular–path)
equilibria. Such a restriction would imply the permanent play of(d,d), which is neither individually
rational nor Pareto efficient in relation to the paths in non–simple equilibria. Generally, the assump-
tion of simple paths is not as restrictive as in PDs, however, as most payoff allocations of non–simple
paths can be approximated in simple paths.
In this context, let us relate our model of IMGs to that of Romano and Yildirim (2004). They
define an IMG with an exogenously fixed and finite number of stages (T), whereas in our case, the
number of stages played is endogenous (and indefinite). Thus, there is a last period in their model, and
in that last period, the players would play a strategy profile that is a Nash equilibrium within the sets of
remaining strategies (this feature allows the play to be induced backwardly). For instance, the unique
outcome in a Prisoner’s Dilemma would be(d,d), regardless of what the number of rounds is. In our
model, to the contrary, there is no last stage where(d,d) must necessarily be played, which allows the
players to collude—leading to the unique outcome of(c,c) in PDs. This intuition generalizes to all
competitive games: in finite IMGs, the equilibria imply either the Nash outcome of the stage game, a
Stackelberg outcome, or an intermediate outcome (partially Nash, partially Stackelberg), whereas in
typical applications of infinite IMGs the unique equilibrium implies collusion at (or, near) the cartel
strategies.
2.2 Public Good Games
Next, let us consider irreversibilities in infinitely repeated public–good games. In a public–good
game, the players can contribute amounts as they are described in their strategy setsSb = {0,1, . . . ,n},and the payoff ofb is Ub(s1,s2) = δ∗ (s1 +s2)−sb. In this context, a strategysA
b is understood to be
more competitive thansBb if it implies a smaller contribution. In turn, this implies for IMGs that the
players can only deviate from their last stage contribution by decreasing it. For instance, a fisher can
enlarge his fleet, but he can not reduce it.
Apparently, under renegotiation proofness, contributions of(n,n,) must result along the equilib-
rium path. Precisely, when the players have played(1,1) in the previous stage, the situation is equiv-
alent to a PD, and further restrictions are Pareto dominated (and even weakly dominated). Hence,
6
(1,1) is an endpoint, and there is generally a player who would deviate from points as(1, i) with
i > 1. As a result, no player would deviate from restrictions to(2,2), since deviations would trigger
the endpoints(0,0) or (1,1). In this way, the backward induction can be completed to show that
(n,n) is the unique endpoint of equilibrium paths. Thus, Pareto efficiency is generally and uniquely
secured.
To the contrary, in plain repeated games, the set of contributions sustained in renegotiation–proof
equilibria includes all profiles(i,n) with i ≤ n that are individually rational. Thus, the outcomes of
repeated games differ significantly from those of IMGs (under renegotiation proofness) even if we
restrict the attention to simple paths. Since the equilibria of IMGs are payoff–equivalent to the equi-
libria of perturbed repeated games, this shows that introducing possibilities to restrict one’s options in
repeated games can have significant strategical implications. Such refinement effects can be observed
whenever there are several pure strategy profiles in the constituent game that imply Pareto efficient
payoff allocations.
Our model of public good games complements previous studies of irreversibilities in public
good games, e.g. Admati and Perry (1991), Marx and Matthews (2000), Gale (2001), and in the
most general way in Lockwood and Thomas (2002). In these studies, precisely the opposite direc-
tion of irreversibilities had been assumed, i.e. the players had been allowed only to increase their
contributions in the course of the game. The applications of these models include (Lockwood and
Thomas, 2002) donations to funds, pollution reductions, capacity reductions in a declining industry,
disarmament, and tariff reductions. Apart from fund raising, the studied models are understood as
mechanisms to de–escalate conflicts. To the contrary, we study circumstances where the players are
best off preventing the conflict from escalating in the first place.
In relation to the outcomes that result in our models, their outcomes are generally not efficient
(neither Pareto efficient nor socially efficient). Apparently, when players can only reduce their con-
tributions (as in our model), the players are generally best off imitating their opponent’s behavior
(i.e. to reciprocate negatively). However, when the contributions are only to be increased, imitating
the opponents is not generally equilibria. In fact, where one would contribute is largely independent
of how much the opponent has contributed before; mainly, it depends on whether one’s contribution
would trigger further contributions of the opponent. Generally, the latter is not satisfied at the ends
of the game tree, i.e. in subgames with previous contributions of(n−1,n−1), and iteratively, this
can prevent contributions to be made altogether.
2.3 Symmetric Competition
The next examples concern irreversibilities in models of Bertrand and Cournot competition. First,
we shall examine a model of competition where the consumers react symmetrically to decisions of
7
the duopolists. That is, the payoff functions are assumed to satisfy for all playersb
Ub(sb,s−b) = sb∗ (1−sb +αs−b) with |α| ≤ 1. (1)
Notably, if we understand thesb to represent prices, a Bertrand model results, and if they represent
quantities, a Cournot model results. Generally, the strategical issues that arise are independent of the
paradigm that we assume: n a model withα > 0, the strategies are complementary (in the sense of
Bulow et al., 1985), and in a model withα < 0, the strategies are substitutionary. However, ifα > 0
and we understand thesb as prices, the underlying goods are substitutes of the consumers’ eyes, but
if α > 0 and we understand thesb as quantities, the underlying goods are complements. In the case
of quantity competition,α = 1 implies (as a limiting case) the Cournot model of homogenous goods.
The best–reply functions, the Nash strategiessNb ≡ s∗b, and the cartel strategiessM
b are
Rb(s−b) =1+αs−b
2, sN
b =1
2−α, sM
b =1
2−2α. (2)
We see thatsNb < sM
b ⇔ α > 0. Thus, in the case of Bertrand competition andα < 0, the customers
would benefit from tacit collusion (towards the cartel prices), as it implies decreasing prices. Like-
wise, the customers benefit under Cournot competition andα > 0 (as the quantities increase), and
generally, they benefit in cases of complementary goods. Moreover, forα → 1, the cartel strategies
converge to infinity, i.e. the players would collude at nearly infinite prices or quantities.
Again, we define IMGs as games where adaptations of the stage game strategies are possible only
towards more competitive strategies. In this case, the relation “more competitive” is defined through
the relation of Nash and cartel strategies, with the Nash strategies being competitive and the cartel
strategies being collusive. Thus, ifsNb < sM
b , then a strategysAb is more competitive thansB
b iff sAb < sB
b.
As a result, the strategies may only be decremented in the course of the interaction. Alternatively, if
sNb > sM
b , then the strategies may only be incremented. Note that in the case of complementary goods,
these directions are somewhat unintuitive (increasing prices, decreasing quantities), but generally, we
would assume that competitors produce rather substitutionary than complementary goods.
Since our strategy sets are assumed to be discrete, all strategies can be represented asskb = sN
b +kεfor some sufficiently smallε > 0. The cartel prices correspond withK = ε−1∗α
2(1−α)(2−α) and k = 0
represents the competitive (Nash) strategies. Apparently, ifα > 0 thenK > 0 and if α < 0 then
K < 0. To simplify the notation, let us define the range(0, . . . ,K) to be understood as(K, . . . ,0) in
caseK < 0.
Our backward induction of the play in the irreversible move game starts ink = 0 and ends
in k = K (which is the opposite direction of the actual irreversibilities). To simplify the backward
induction, let us assume in this section that all strategiesskb that can be defined through ak∈ (0, . . . ,K)
are indeed in playerb’s strategy set. Thus, the following identity holds for both playersb∈ B.{sb ∈ Sb : sN
Nash equilibria in all subgames (Bernheim et al., 1987).
Additionally, let us impose an assumption that simplifies some of the notation but is irrelevant oth-
erwise. In particular, it does not affect the equilibrium payoffs, i.e. we could assume precisely the
opposite direction without affecting the equilibrium payoffs.
15
Assumption 3.6 If two endpoints induce equivalent payoffs and if one of them is less competitive
than the other one, the equilibrium path implies the more competitive one.
3.3 Characteristics of Pareto–Perfect Equilibria (PPEs) in IMGs
In order to get a basis for backward inductions in IMGs, we shall first determine the endpoints that
are not below the Nash equilibriums∗. These endpoints are described in the following Lemma (its
proof is relegated to the appendix).
Lemma 3.7 Consider an arbitrary PPE ˆs of an IMG. A points≮ s∗ is an endpoints∈ E induced in
s if and only if s≥ R(s). The endpoints∗ Pareto dominates all endpointss′ ≥ s∗.
Thanks to Lemma 3.7, we know all endpointse that satisfye� s∗. In the following, we shall
backward induce the remaining endpoints.
Proposition 3.8 In any PPE of an IMG, the induced set of potential endpointsE satisfies for alls∈S
thats∈ E if and only if s Pareto dominates all points inEs = {e∈ E : e> s}. Thus,E is unique.
PROOF. The “if” part of the proposition follows directly from the characteristics of Pareto–perfect
equilibria. To prove the “only if” part, we shall contradict the opposite hypothesis. Assume there is a
PPE such that there are endpointss∈E that do not Pareto dominate all points inEs = {e∈E : e> s}.Let Econ denote the set of alls∈ E that contradict our claim, and consider anysc ∈ maxEcon. As a
result, the above proposition must be satisfied for alls′ > sc. Since it is not satisfied forsc, however,
there is somee∈ E : e> sc that is not Pareto dominated bysc, and hence, there is a playerb such that
Ub(e) > Ub(sc).
Whenb deviates unilaterally toeb, one of the Pareto efficient endpoints inE′ = {e′ ∈ E : e′ ≥(eb,sc
−b)} would result. Generally, playerb is better off in(eb,sc−b) than ine, assc
−b < e−b. All of the
Pareto efficient points ine′ 6= e∈ E′ satisfy (by assumption) thate′ ≯ e, and thuse′−b < e−b. Thus,
in any continuation equilibrium following a stage with(eb,sc−b), b’s opponent plays a stage strategy
s−b ≤ e−b. Hence, no such continuation equilibrium can imply a payoff allocation whereb is worse
off than ine, asb could deviate unilaterally to a strategy where he does not move beyondeb. As a
result,b can secure the payoff resulting ine, and ansc must not be an endpoint. QED
Corollary 3.9 There is a unique set of endpointsE that is induced in all PPEs of a given IMG.
The uniqueness of the set of endpoints does not generally imply that the equilibrium paths in all
subgames are unique. Consider a stagek ∈ KD with st = s, define the set of endpoints that can still
be reached asEs{e∈ E : e≥ s} . Under renegotiation proofness, the continuation equilibrium must
lead to one of the Pareto efficient points inEs. Thanks to the construction ofE (and thusEs), this
implies that the subgame’s equilibrium path ends in one of the points in minEs.
16
Lemma 3.10 Consider a decision stagek ∈ KD and denote the set of potential endpoints in the
resulting subgame asEst = {e∈ E : e≥ st}. Any PPE path in that subgame must imply an endpoint
in minEst .
Thus, a subgame’s equilibrium path is unique (apart from payoff–equivalence and interchangeability)
iff minimum of Est is unique. The uniqueness is examined in the following. First, consider the
following lemma.
Lemma 3.11 For all pairs of endpointse1 6= e2 ∈E : e1,e2 < s∗, eithere1 > e2 or e2 > e1 is satisfied.
Moreover, all endpointse< s∗ are interior, i.e. they satisfye≤ R(e). (The proofs are given in the
appendix.)
As a result, the set of endpoints that are less competitive thans∗, E′ := E∩{s∈ S : s≤ s∗}, can
be completely ordered by “�”. That is, we can connect all of these endpoints through a (unique)
monotonically increasing line (see the examples in Section 2). In some sense, this line characterizes
the strategical aspects of IMGs. This leads straightforwardly to the following.
Lemma 3.12 There is generally a unique endpoint that can result along the path of a PPE where
no player is worse off than in the constituent game’s Nash equilibriums∗ (the proof is given in the
appendix).
As a result, if for all endpointse∈E : e≮ s∗ we havee≥ s∗ (as in the case of strategic complements),
the endpoint that results along the equilibrium path is unique, and no player is worse off than ins∗.
Next, let us illustrate the relative equilibrium payoffs if the Pareto efficient endpoint is not unique.
Lemma 3.13 Let EPs denote the set of potential Pareto–efficient endpoints in a stagek ∈ KD with
st = s. For alle′ 6= e′′ ∈ EPs , we have for allb thate′b > e′′b ⇔ Ub(e′) > Ub(e′′) (the proof is given in
the appendix).
This observation allows us to establish the payoff–equivalence of IMGs and generalized/perturbed
repeated games (see below). In the following, we shall use it to illustrate the assumption of alternating
moves in IMGs, instead of Pareto perfectness. Here, we shall assume that for alls′ 6= s′′ ∈ S in the
constituent game and for all playersb, the induced payoffs are differentUb(s′) 6=Ub(s′′). Generically,
this is satisfied. Thus, it is straightforward to induce (backwardly) that the sets of potential endpoint
E is equivalent to that under Pareto perfectness and that Pareto–efficient paths result in all subgames.
More significantly, as a consequence of Lemma 3.13 we have for all playersb, all s∈ S, and
all stagesk∈ KD : st = s, that if e∈ Es denotes the (necessarily unique) endpoint that maximizesb’s
payoff, thene is also the unique Pareto efficient endpoint in a subgame following the play(eb,s−b).
As a result, the first moving player in a alternating–move IMG can secure the endpoint that maximizes
his payoff, and thus, the equilibrium outcome is unique.
17
Proposition 3.14 Generically, in IMGs with alternating moves, the subgame–perfect equilibrium
outcome (payoff allocation) is unique. (Note that we need not assume Pareto perfectness.)
3.4 Simply Structured Competitive Games
If the stage game is (beyond its competitiveness) simply structured in the sense defined below, the
endpoints are particularly easy to induce. All of the examples that we gave initially are simply
structured in this sense. The simplified way to induced the endpoints is defined first.
Definition 3.15 Simplified Induction Start withs0 = s∗ and construct the pointssk+1 = max{s∈S : s� sk} until the newly constructed point fails to yield Pareto improvements. The last point
implying Pareto improvements is calledsK.
Thus, we construct a sequence of pointssK < · · · < sk < sk−1 < · · · < s1 < s∗ which is connected in
the sense that the deviation from somesk to the next pointsk−1 requires precisely one step per player
(towards a more competitive strategy). This is simplified, we can not go back simply one step per
player in general. Instead, we have to look for the “next” point that Pareto dominates all endpoints
that have already been found.
The set of simply structured competitive games is defined such that all endpointse< s∗ can be
derived in the simplified induction if and only if the IMG is based on a simply structured game. Note
that the remaining endpointse≮ s∗ are described in Lemma 3.7. Thus, when we have shown that a
game is simply structured, all endpoints can be derived easily.
Definition 3.16 Simply structured competitive gameA simply structured competitive game is a
competitive game that satisfies for the(sk)Kk=0 constructed in the stepwise induction
∀k : sk ≤ R(
sk)
(14)
∀k′ < k′′ : U(
sk′)
< U(
sk′′)
(15)
∀b, ∀k, ∀sb > s∗b :(
sb,sk−b
)≥ R
(sb,s
k−b
)⇒ Ub
(sb,s
k−b
)≤Ub
(sk)
. (16)
∀s< sK : Ub(s) ≯ Ub(sK) (17)
∀s : s≥ R(s) ands≯ sK : U(s) < U(sK) (18)
Verbally, in simply structured competitive games, the path fromsK to s∗ lies between the best reply
curves (14), symmetric deviations to more competitive strategy profiles are Pareto dominated (15),
deviations to endpointse≮ s∗ are generally not profitable (16), the players would generally deviate
from s < sK to sK (17), andsK Pareto dominates all potential endpointss≥ R(s) that can not be
reached anymore whensK is reached (18).
18
Proposition 3.17 The “simplified induction” yields the (unique) set of endpoints satisfyinge≤ s∗ if
and only if the stage game is simply structured competitive. Along any equilibrium path,sK results
(the proof is skipped).
4 Dynamic Games that Are Payoff–Equivalent to IMGs
4.1 Infinite Irreversible Move Games
In infinite irreversible move games (IIMGs), stage game strategy adaptations are irreversible as in
IMGs, but the game never stops and all rounds are payoff–relevant. The set of stages in IIMGs is
K ={(s0,s1, . . . ,st) : t ≥ 0, s0 = minS, ∀t ′ ≤ t : st ′ ∈ Sandst ′ ≤ tst ′−1
}. (19)
There are no terminal stages. An IIMG strategy ˆsb maps each stage to a feasible move, i.e. to some
s∈ {s′ ∈ S: s′ ≥ st}. The path of play followingk∈ K according to ˆs= (sb) is (recursively)