Conceptos básicos de HSDPA

In the previous sections, we have established that it is possible to define extensive form games modeling continuous-time problems without the re- course to an artificially constrained strategy set. It is, however, natural to ask whether there is a relation between the Action-Reaction Framework and previous approaches which employed strategy constraints. Indeed, it is pos- sible to embody ideas similar to the ones in the Action-Reaction Framework through strategy constraints. In this section we detail this alternative route and show how these constraints must be imposed to preserve equivalence (in a well-defined sense to be detailed below) with the extensive form approach. Informally, a Conditional Response Mapping is a mapping which specifies, at each time t, an action (depending only on the previous history of play) and a response which depends on the actions being simultaneously decided by other players. A number of additional conditions must be imposed in order to capture the constraints which are also inherent in the Action-Reaction Framework. Naturally, these additional conditions resemble the restrictions imposed on strategies by e.g. Stinchcombe (1992) and Bergin (2006), among others (see Section 1.6). The reason we refrain from using the term strategy is that a priori it is not clear whether the set of Conditional Response Mappings indeed corresponds to the set of strategies in a well-defined extensive form. We shall, however, see that this is the case.

Analogously to the conditions discussed for extensive forms, a coherent framework will be obtained if every profile of mappings induces an outcome contained in the appropriate outcome set and any outcome can be reached by some profile. In order to guarantee these properties, however, it is not sufficient to place restrictions on Conditional Response Mappings only. It is necessary to also constrain the set of possible outcomes, and hence (through the dependence on histories) the domain of these mappings. The appropriate constraints for the set of outcomes are exactly as in the Action-Reaction Framework: outcomes must define decision paths.

Let F denote the set of decision paths as introduced in Definition 3. The formal definition of Conditional Response Mappings is as follows.

Definition 6. A Conditional Response Mapping (CRM) for player i ∈ I is a mapping σi : F × R+→ A2i, (f, t) 7→ (σi1(f, t), σi2(f, t)) such that for every

f ∈ F and all t ∈ R+

(CRM.i) if f (τ ) = f′_{(τ ) for f}′ _{∈ F and all τ ∈ [0, t[, then σ}1

i(f, t) = σi1(f′, t);7

if f (τ ) = f′_{(τ ) for f}′ _{∈ F and all τ ∈ [0, t], then σ}2

i(f, t) = σi2(f′, t).

(CRM.ii) if t ∈T

j∈ILC(fj)



∪ J(fi) then σi2(f, t) = fi(t) and there is εi(f, t) >

0 such that σ1

i(f, τ ) = fi(t) for all τ ∈]t, t + εi(f, t)[.

(CRM.iii) if t ∈ LC(fi) ∩ S_k∈IJ(fk)

then there is εi(f, t) > 0 such that

σ1

i(f, τ ) = fi+(t) for all τ ∈]t, t + εi(f, t)[.

Denote the set of CRMs for player i by Σi and let Σ := ×i∈IΣi.

For each decision path f and time t, a CRM hence specifies an action, denoted σ1

i(f, t), and an instant response σ2i(f, t). The first part of condition

(CRM.i) specifies that actions depend only on the past history of play, i.e. on the values of f up to (but excluding) t. The second part of this condition stipulates that responses depend only on the values of f up to and including t. Equivalently, at time t each player specifies an action and, for any possible profile of actions at t which is part of a decision path, also a conditional response.

Condition (CRM.ii) captures the intuition that, as long as no player has changed action at t (and hence the decision path is left-continuous in all coordinates), then no player can change the current action through a condi- tional response. That is, “no reaction without a triggering action”. Further, players will be constrained to the current action for some small time interval. Likewise, the same restrictions apply if a given player has changed action at t (“jumped”), which embodies the intuition that two action changes of a given player cannot be arbitrarily close. In particular, all players have to stick to the action picked at time 0 for some positive amount of time.

7

Condition (CRM.iii) captures a similar intuition for responses. If a player did not initiate an action change at t, but some other player did, then the original player was allowed to react through the stipulated conditional re- sponse (hence no constraint is placed on the second component of the action tuple). The condition requires that the player needs to stick to the action specified as a response for some small time interval, as long as no other player initiates an action change.8

The following examples illustrate that the restriction to decision paths is necessary. In other words, CRMs are well-defined mappings only on F × R+. The first example shows that a CRM cannot be built by simply gluing

together arbitrary chains of decisions.

Example 1. Let I = {1} and A1 = {0, 1} and consider the function h : R+→

A1 defined by h(τ ) =    0, if τ ∈ Q, 1, if τ ∈ R \ Q.

By (CRM.ii), for any t ∈ Q, there should exist an ε > 0 such that σ1_{(h, τ ) = 0}

for all τ ∈]t, t + ε[. Also by (CRM.ii) for any t′ _{∈ R \ Q∩]t, t + ε[ there should}

exist an ε′ _{> 0 such that σ}1_{(h, τ ) = 1 for all τ ∈]t}′_{, t}′ _{+ ε}′_{[, which leads to a}

contradiction.

In this example, the decision maker changes action “too often”, with action changes being arbitrarily close to each other. The next example shows that this problem also arises with more intuitive, “continuous” mappings.

Example 2. Let I = {1} and A1 = R+. Consider the function h : R+ → A1

defined by h(τ ) = τ for all τ ∈ R+. By (CRM.ii) for any t ∈ R+, there should

exist an ε > 0 such that σ1_{(h, τ ) = t for all τ ∈]t, t + ε[. Also by (CRM.ii)}

for any t′ _{∈]t, t + ε[ there should exist an ε}′ _{> 0 such that σ}1_{(h, τ ) = t}′ _{> t}

for all τ ∈]t′_{, t}′_{+ ε}′_{[, which leads to a contradiction.}

The last two conditions in the definition of CRM implicitly incorporate a notion of “inertia” analogous to the inertia times needed for the Action- Reaction Framework. A first step in order to show that the new framework

8

is coherent is to make the relationship to inertia times explicit. This corre- sponds to the following thought experiment. Given a decision path f and a time instant t, imagine the path after t was changed in such a way that nobody changed action after the reactions specified at time t, i.e. the path was fixed at f+(t). What is the first point in time after t such that a given

CRM σi would specify a deviation from the new path? If such a first point

in time is well defined and equal to t + ε, the quantity ε will fulfill the same role as an inertia time in the Action-Reaction Framework.

Let us formally construct these inertia times for a given CRM σi ∈ Σi.

For each f ∈ F and t ∈ R+let ft+ be given by ft+(τ ) = f (τ ) for all τ ≤ t and

ft+_{(τ ) = f}

+(t) for all τ > t. We call ε > 0 a deviation point prescribed by σi

after (f, t) if σ1

i(ft+, t+ε) 6= fi+(t). That is, for a deviation point ε, the action

prescribed by σi at time t + ε is different from the action/reaction chosen

by player i at time t, given that all players stick to their actions/reactions chosen at time t. The following lemma shows that whenever such a deviation point exists, there is a first deviation point, which then plays the role of an inertia time.

Lemma 2. Let f ∈ F , t ∈ R+, i ∈ I, σi ∈ Σi, and let Eσi(f, t) be the set of

all deviation points prescribed by σi after (f, t). If Eσi(f, t) 6= ∅ then there

exists a first deviation point εσi(f, t) = min Eσi(f, t).

This property is a consequence of condition (CRM.ii). It should be remarked that the existence of a first deviation point corresponds to the “Identifiability” assumption for admissible strategies imposed by Stinchcombe (1992). The difference is that in Stinchcombe (1992), this property is imposed as one of the conditions restricting the strategy set, while in our framework, it is a property derived from the definition of CRM.

The existence of first deviation points as identified in Lemma 2 is crucial for the framework at hand. It has two important consequences. First, it plays a major role in the proof of outcome existence and uniqueness below. Second, and as already announced, the εσi(f, t) essentially reconstruct inertia times

and will allow us to establish the equivalence between the Action-Reaction Framework and the framework based on CRMs.

The next result shows that this framework is coherent, that is, every profile of CRMs induces a unique outcome (after every history) and any outcome in F can be reached by some profile of CRMs.

Definition 7. Let σ ∈ Σ.

(i) f ∈ F is induced by σ if σ1

i(f, t) = fi(t) and σ2i(f, t) = fi+(t) for all

i ∈ I and all t ∈ R+.

(ii) Given f ∈ F and t ∈ R++, f ∈ F is induced by σ after (f, t) if

f (τ ) = f (τ ) for all τ ∈ [0, t[, σ1

i(f, τ ) = f (τ ) and σi2(f, τ ) = fi+(τ ) for

all τ ∈ [t, +∞[ and i ∈ I.

Proposition 2. (i) Every σ ∈ Σ induces a unique f ∈ F .

(ii) For all f ∈ F and t ∈ R++ every σ ∈ Σ induces a unique f ∈ F after

(f , t).

(iii) Every f ∈ F is induced by some σ ∈ Σ.

Properties (i) and (ii) in the last Proposition are comparable to Theorem IV.1 in Stinchcombe (1992), Theorem 3 in Bergin (1992), Lemma A.1 in Perry and Reny (1993), Theorem 2 in Bergin and MacLeod (1993), and Theorem 1 in Bergin (2006). All these results state that, under the constraints of the respective framework, every profile of strategies induces a unique outcome after any history. Property (iii) additionally states that any outcome can be reached by some profile of strategies, a result similar to Theorem IV.2 in Stinchcombe (1992).

The intuition behind the proof of the last result is as follows. Given a profile of CRMs, initial actions are clear. The inertia times identified in Lemma 2 then allow us to identify the (constant) path up to the next deviation point. At that point, the CRMs can be used to establish the new actions/reactions. Applying Lemma 2, the construction can be iterated. Since time is continuous, the exact iterative argument relies on transfinite recursion, which is made possible by the structure of decision paths.