Non-reactive strategies in decision-form games

Non-reactive strategies in decision-form games

David Carfì

Faculty of Economics, University of Messina Email: [email protected]

Abstract In this paper we propose a concept of rationalizable solution for two-player decision-form games: the solution by iterated elimination of non-reactive strategies. Several original theorems are proved about this kind of solution. We study the relations between solutions by iterated elimination of non reactive strategies and game equilibria. We present an existence theorem for bistrategies surviving the iterated elimination and an existence theorem for solution by iterated elimination in contracting games. We, also, show that an equilibrium of a game survives iterated elimination of non reative strategies. At the end we prove a characterization of solvability by iterated elimination of non-reactive strategies. 1

Introduction

The concept of solution by iterated elimination of non-reactive strategies, for two-player decision-form games was conceived by the author and presented in [27] and [47], there (in [27]) two-player decision-form games were introduced. In this paper decision rules are used in the sense introduced by J. P. Aubin in [2] and [1], and they represent the action-rationality, the behavioural way itself, of each player in front of the competitive situation represented by the game. For different concepts of rationalizable solution, for instance solutions obtained by elimination of dominated strategies, the reader can see in [49],

The context.

[50] and [48].

consider two non-void sets

The aim is to form ordered pairs of strategies

called strategy profiles or bistrategies, via the (individual or

𝑥 and 𝑦 , done by the two players in 𝐹 , respectively, in order that the strategy 𝑥 of the first player

collective) selection of their components the sets

𝐸

and

We shall

𝐸 and 𝐹 , viewed as the respective sets of strategies

at disposal of two players.

(𝑥, 𝑦) ∈ 𝐸 × 𝐹 ,

We deal with two-player games.

1

is a good reaction to the strategic behavior

𝑦

of the second player and vice

versa.

Definition (strategy base and bistrategy space). Let (𝐸, 𝐹 ) be a pair of non-empty sets, we call it strategy base of a two-player game. The first set 𝐸 is said the first player’s strategy set; the second set 𝐹 is said the second player’s strategy set. Any element 𝑥 of 𝐸 is said a first player’s strategy and any element 𝑦 in 𝐹 is said a second player’s strategy. Every pair of strategies (𝑥, 𝑦) ∈ 𝐸 ×𝐹 is said a bistrategy of the strategy base (𝐸, 𝐹 ) and the cartesian product 𝐸×𝐹 is said the bistrategy space of the base (𝐸, 𝐹 ). Interpretation and terminology. We call the two players of a game Emil and Frances: Emil, simply, stands for “first player”; Frances stands for

𝐸 , Frances’ aim is to choose a strategy 𝑦 in 𝐹 , in order to form a bistrategy (𝑥, 𝑦) such that the strategy 𝑥 is an Emil’s good response to the Frances’ strategy 𝑦 and

“second player”. Emil’s aim is to choose a strategy

𝑥

in the set

vice versa.

Definition (decision rule). Let (𝐸, 𝐹 ) be a strategy base of a twoEmil’s decision rule on the base (𝐸, 𝐹 ) is a correspondence from 𝐹 to 𝐸 , say 𝑒 : 𝐹 → 𝐸. Symmetrically, a Frances’ decision rule on the base (𝐸, 𝐹 ) is a correspondence from 𝐸 to 𝐹 , say 𝑓 : 𝐸 → 𝐹 . Definition (decision-form game). Let (𝐸, 𝐹 ) be a strategy base of a two-player game. A two-player decision-form game on the base (𝐸, 𝐹 ) player game. An

(𝑒, 𝑓 ) of decision rules of the players Emil and Frances, strategy base (𝐸, 𝐹 ).

is a pair on the

respectively,

Definition (of possible reaction and of capability of reaction).

Let

(𝑒, 𝑓 )

be a decision-form game.

ments of the image of the set

𝑒(𝑦)),

𝑦

Let

𝑦

be a Frances’ strategy, the ele-

𝑒

by the correspondence

(that is, the elements of

i.e., the direct corresponding strategies of

𝑦

by the rule

𝑒,

are

Emil’s possible responses, or Emil’s possible reactions, to the Frances’ strategy 𝑦. Analogously, let 𝑥 be an Emil’s strategy, the elements called

of the image of

𝑓 (𝑥)),

𝑥

by the decision rule

𝑓

(that is, the elements of the set

i.e. the direct corresponding strategies of

𝑥

by the rule

𝑓,

are said

Frances’ possible responses, or Frances’ possible reactions, to the Emil’s strategy 𝑥. The set of Emil’s possible reactions (responses) to the Frances’ strategy 𝑦 is said the Emil’s reaction set to the Frances’ strategy 𝑦. Finally, we say that Emil can react to the Frances’ strategy 𝑦 if the corresponding reaction set

𝑒(𝑦)

is non-void.

Definition (of a disarming strategy).

Emil’s strategies

𝑓 (𝑥)

𝑥

Let

(𝑒, 𝑓 )

be a game.

The

to which Frances cannot react, i.e. such that the image

is empty, are called

Emil’s disarming strategies (for Frances). 2

The Frances’ strategies reaction set

𝑒(𝑦)

𝑦

to which Emil cannot react, namely such that the

is empty, are called

Emil). Definition (of subgame).

Frances’ disarming strategies (for

Let 𝐺 = (𝑒, 𝑓 ) be a decision-form game with (𝐸, 𝐹 ) and let (𝐸 ′ , 𝐹 ′ ) be a subbase of (𝐸, 𝐹 ), namely a pair of subsets of 𝐸 and 𝐹 , respectively. We call subgame of 𝐺 with strategy base (𝐸 ′ , 𝐹 ′ ) the pair (𝑒′ , 𝑓 ′ ) of the restrictions of the decision rules 𝑒 and 𝑓 to the pairs of sets (𝐹 ′ , 𝐸 ′ ) and (𝐸 ′ , 𝐹 ′ ), respectively. It is important to ′ ′ ′ remember that 𝑒 is the correspondence from 𝐹 to 𝐸 which associates with ′ ′ ′ ′ every strategy 𝑦 in 𝐹 the part 𝑒(𝑦 ) ∩ 𝐸 . In other words, it sends every ′ ′ ′ strategy 𝑦 of 𝐹 into the corresponding Emil’s reaction strategies to 𝑦 which ′ ′ ′ belong to 𝐸 . We also call the subgame (𝑒 , 𝑓 ) the restriction of the game 𝐺 to the strategy pair (𝐸 ′ , 𝐹 ′ ).

strategy base

2

Reactive strategies

In a decision-form game, if a certain player’s strategy any strategy of the other one, this strategy

𝑠

𝑠

does not react to

can’t be a reasonable action of

the first player. For this reason, we are motivated to formalize, in the below definition, the concept of non-reactive strategy.

Definition (of a reactive strategy).

Let

(𝑒, 𝑓 ) be a two-player decision-

reactive (with respect to the decision rule 𝑓 ) if it is a possible reaction to some Emil’s strategy. form game. Let

𝑦0

be a Frances’ strategy, we call it

In other words, a Frances’ strategy if it belongs to the set

𝑓 (𝑥),

𝑦0

is called reactive (with respect to

for some Emil’s strategy

𝑥.

𝑓 ),

A Frances’ strategy

non-reactive if it is not reactive. Analogously, let 𝑥0 be an Emil’s strategy, we call it reactive (with respect to the decision rule 𝑒) if it is called

is a possible reaction to some Frances’ strategy. In other words, an Emil’s strategy

𝑥0

is called reactive (with respect to

for some Frances’ strategy

𝑦.

𝑒),

if it belongs to the set

An Emil’s strategy is called

𝑒(𝑦),

non-reactive if it

is not reactive.

Remark (on the sets of reactive strategies).

Emil’s and Frances’

∪𝑒 :=

⋃︀

𝑒(𝑦) and ∪𝑓 := 𝑥∈𝐸 𝑓 (𝑥), i.e., the images of the correspondences 𝑒 and 𝑓 , respectively. Note that, for example, with the correspondence 𝑒 : 𝐹 → 𝐸 it is, in a standard way, associated with the mapping 𝑀𝑒 : 𝐹 → 𝒫(𝐸), sending any Frances’ strategy 𝑦 into the reaction set 𝑒(𝑦). The mapping 𝑀𝑒 is, therefore, a family of subsets of 𝐸 indexed by the set 𝐹 . Analogously, for the correspondence 𝑓 , we can consider the family 𝑀𝑓 = (𝑓 (𝑥))𝑥∈𝐸 . So, the sets of respective reactive strategies are the two unions

⋃︀

3

𝑦∈𝐹

above two unions are the unions of the families

3

𝑀𝑒

and

𝑀𝑓 ,

respectively.

Elimination of non-reactive strategies

Definition (of a reduced game by elimination of non reactive strategies). A game (𝑒, 𝑓 ) is called reduced by elimination of non-reactive strategies if the images of the decision rules 𝑒 and 𝑓 are the strategy sets 𝐸 and

𝐹,

respectively. In other words, the game is reduced if the decision rules

of the two players are onto. In a game, a rational behavior of the players is to use only reactive strategies, eliminating the non-reactive ones. So, they will play a subgame of the previous one, that we call reduction of the game by elimination of non-reactive strategies. that, if

𝐹 :𝑋→𝑌

Before defining the reduction of a game we recall

is a correspondence and if

𝑋′

𝑌 ′ are subset of 𝑋 and 𝐹 is the correspondence

and

𝑌, respectively, the restriction to the pair (𝑋 ′ , 𝑌 ′ ) of 𝐹|(𝑋 ′ ,𝑌 ′ ) whose graph is gr(𝐹 ) ∩ (𝑋 ′ × 𝑌 ′ ).

Definition (the reduction of a game by elimination of nonreactive strategies). Let (𝑒, 𝑓 ) be a decision-form game on the strategy base (𝐸, 𝐹 ). We call (first) reduction of the game (𝑒, 𝑓 ) by elimination of non-reactive strategies the subgame (𝑒′ , 𝑓 ′ ) on the subbase (𝑒 (𝐹 ) , 𝑓 (𝐸)), pair of images of the decision rules

𝑒

and

𝑓,

respectively. In other words, the

(first) reduction of the game (𝑒, 𝑓 ) by elimination of non-reactive strategies is the game whose decision rules are the restrictions 𝑒|(𝐹 ′ ,𝐸 ′ ) and 𝑓|(𝐸 ′ ,𝐹 ′ ) ,

where

𝐸′

and

𝐹′

are the images of the rules

𝑒

and

𝑓.

As we saw, the first reduction of a game can be non-reduced, so, we can consider the successive reductions to find a reduced subgame.

Definition (of 𝑘-th reduction by elimination of non-reactive strategies).Let 𝐺0 = (𝑒0 , 𝑓0 ) be a game on a strategy base (𝐸0 , 𝐹0 ) and let 𝑘 be a natural number. We define (recursively) the 𝑘 -th reduction, or reduction of order 𝑘, by elimination of non-reactive strategies of the game 𝐺0 as follows: the same game 𝐺0 , if 𝑘 = 0; the subgame 𝐺𝑘 = (𝑒𝑘 , 𝑓𝑘 ) (𝑘 − 1) 𝑒𝑘−1 (𝐹𝑘−1 ) and 𝑓𝑘−1 (𝐸𝑘−1 ), if 𝑘 ≥ 1. In other words, if 𝑘 ≥ 1, the decision rules 𝑒𝑘 and 𝑓𝑘 are the restrictions to the pairs (𝐹𝑘 , 𝐸𝑘 ) and (𝐸𝑘 , 𝐹𝑘 ) of the decision rules 𝑒𝑘−1 and 𝑓𝑘−1 , respectively. We say that a strategy 𝑥0 ∈ 𝐸 survives the 𝑘 -th elimination of non-reactive strategies if it belongs to 𝐸𝑘 .

on the base

(𝐸𝑘 , 𝐹𝑘 ),

pair of the images of the decision rules of the

reduction, i.e., pair of the sets

Theorem (on the values of the reduced decision rules).

conditions of the above definition, for each strategy

4

𝑠

In the

of a player, which

survived the

𝑘 -th

elimination of non-reactive strategies, the reaction set of

the other player remains unchanged. In particular, if the game disarming strategies, all the reductions

𝐺𝑘

𝐺0

has not

has not disarming strategies.

Definition (reducing sequence by elimination of non-reactive strategies). Let 𝐺0 = (𝑒0 , 𝑓0 ) be a game on a strategy base (𝐸0 , 𝐹0 ). We define reducing sequence by elimination of non-reactive strategies of the game 𝐺0 the sequence of reduced subgames 𝐺 = (𝐺𝑘 )∞ 𝑘=0 . In other words, it is the sequence with first term the game term the

𝑘 -th

reduction of the game

𝐺0

itself and with

𝑘 -th

𝐺0 .

The reducing sequence allows us to introduce the concept of solvability and solution by iterated elimination of non-reactive strategies.

Definition (of solvability by iterated elimination of non-reactive strategies). Let 𝐺0 = (𝑒0 , 𝑓0 ) be a decision-form game and let 𝐺 be its reducing sequence by elimination of non-reactive strategies. The game called

𝐺0

solvable by iterated elimination of non-reactive strategies

is if

there exists only one bistrategy common to all the subgames of the sequence

In this case, that bistrategy is called solution of the game 𝐺0 by iterated elimination of non-reactive strategies. Remark. The definition of solvability by iterated elimination of non-

𝐺.

reactive strategies means that the intersection of the bistrategy spaces of all the subgames forming the reducing sequence, that is the intersection

⋂︀∞

𝑘=0 𝐸𝑘

× 𝐹𝑘 ,

has one and only one element, which we said the solution of

the game.

Remark.

If the game

𝐺0

is finite, it is solvable by iterated elimination of

non-reactive strategies if and only if there exists a subgame of the sequence

𝐺

with a unique bistrategy. In this case, that bistrategy is the solution, by

iterated elimination of non-reactive strategies, of the game

4

𝐺0 .

Iterated elimination in Cournot game

The game.

𝐺0 = (𝑒0 , 𝑓0 ) be the Cournot decision-form game with 2 bistrategy space the square [0, 1] and (univocal) decision rules defined, for ′ every 𝑥, 𝑦 ∈ [0, 1], by 𝑒0 (𝑦) = (1 − 𝑦)/2 and 𝑓0 (𝑥) = (1 − 𝑥)/2. Set 𝑥 = 1 − 𝑥 ′ and 𝑦 = 1 − 𝑦 , the complement to 1 of the production strategies, for all ′ ′ bistrategy (𝑥, 𝑦) of the game; briefly, we have 𝑒0 (𝑦) = 𝑦 /2 and 𝑓0 (𝑥) = 𝑥 /2. ∞ The reducing sequence. Let 𝐺 = (𝐺𝑘 )𝑘=0 be the reducing sequence by elimination of non-reactive strategies of the game 𝐺0 . The sequence 𝐺 has as starting element the game 𝐺0 itself. Let 𝐸𝑘 and 𝐹𝑘 be the strategy spaces of the 𝑘 -th game 𝐺𝑘 = (𝑒𝑘 , 𝑓𝑘 ), for all natural 𝑘 . The base of the game 𝐺𝑘+1 Let

5

is, by definition, the pair of images

strategy spaces.

The function

𝑒𝑘

𝑒𝑘 (𝐹𝑘 )

and

𝑓𝑘 (𝐸𝑘 ).

Reduction of the

is strictly decreasing and continuous, so,

[𝑎, 𝑏] is the compact interval [𝑒𝑘 (𝑏), 𝑒𝑘 (𝑎)]. The initial strategy spaces 𝐸0 and 𝐹0 are intervals, then, by induction, all the spaces 𝐸𝑘 = 𝐹𝑘 are intervals. Let [𝑎𝑘 , 𝑏𝑘 ] be the 𝑘 -th strategy space 𝐸𝑘 , then, con′ ′ ′ ′ cerning the (𝑘 + 1)-th, we have 𝐹𝑘+1 = [𝑏𝑘 /2, 𝑎𝑘 /2] . The interval [𝑏𝑘 /2, 𝑎𝑘 /2] does not coincide with the space 𝐹𝑘 = 𝐸𝑘 : the reduction 𝐺𝑘 is not reduced. the image of a real interval

Iterated reduction.

We now study the sequence of the initial end-points

([𝑎𝑘 , 𝑏𝑘 ])∞ 𝑘=1 . We have 𝑎𝑘+1 = 𝑒(𝑏𝑘 ) = 𝑒(𝑓 (𝑎𝑘−1 )). The composite function 𝑒 ∘ 𝑓 is increasing and, moreover, the following inequalities 𝑎0 ≤ 𝑎1 ≤ 𝑎2 hold true. So, the sequence 𝑎 is increasing, and, then, it is 1+𝑎𝑘−1 , convergent (as it is upper bounded by 𝑏0 ). Moreover, being 𝑎𝑘+1 = 4 * * * * and putting 𝑎 := lim(𝑎), we deduce 4𝑎 = 1 + 𝑎 , which gives 𝑎 = 1/3. Similarly, we prove that 𝑏 is a decreasing sequence and it converges to 1/3. of the interval family

Solution.

Concluding, the unique point common to all the strategy inter-

⋂︀∞

𝑘=0 [𝑎𝑘 , 𝑏𝑘 ] = {1/3}. Then, the game is solvable by elimination of non-reactive strategies and the solution vals is the strategy

is the bistrategy

1/3, in other terms we have

(1/3, 1/3).

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