Sequential Games with Finite Horizon and Turn Selection Process

Finite Strategy Sets Case

Rub´en Becerril-Borja and Ra´ul Montes-de-Oca

Departamento de Matem´aticas, Universidad Aut´onoma Metropolitana-Iztapalapa,

Av. San Rafael Atlixco 186, Col. Vicentina, 09340, M´exico D.F., M´exico

Keywords:

Sequential Games, Stochastic Games, Turn Selection Process, Existence of Nash Equilibria.

Abstract:

A class of models of sequential games is proposed where the turns of decision are random for all players. The

models presented show different variations in this class of games. In spite of the random nature of the turn

selection process, ﬁrst the number of turns per player is ﬁxed, and afterwards models without this property are

considered, as well as some that allow changes in other components. For all the models, a series of results are

obtained to guarantee the existence of Nash equilibria. A possible application is shown for drafting athletes in

sports leagues.

1 INTRODUCTION

The focus of game theory is on studying situations

in which several individual subjects make decisions

that affect all of them in the form of a utility reward.

Essentially, any interaction that can be roughly de-

scribed as before can be studied as a game.

In the classical theory (Fudenberg and Tirole,

1991), (Tadelis, 2013), the situations that can be anal-

ysed are deterministic in their rules, which means that

a particular structure is established before the players

interact, and such structure has no random elements

once the players are making decisions. That, in itself,

provides a frame to study the game mathematically

and come up with the decisions that have to be taken

in order to maximize the utility of each player, subject

to the fact that each player can only make his own de-

cision.

Another important characteristic to take into ac-

count to study the situation is the time frame of the

game. When the decisions are made in discrete time,

games can be divided in two sets: if players make

their decisions at the same time, or if they only know

about the structure of the game, but are not aware of

the decisions of the other players before making their

own, it is a simultaneous game. If there is a certain

order of turns to make a decision such that players

in the future may know the choices previously made

in order to adapt their behaviour, then we have a se-

quential game. In the latter one, the number of turns

is called the horizon of the game.

The ﬁrst approach to introduce random elements

was made in sequential games, where a player called

Nature was to make its decision before any other

player made his own. Because of the structure of

sequential games, one could decide whether players

would be aware of the behaviour of the Nature player,

which allows to study situations with random exter-

nalities.

Nevertheless, once the game was afoot, there were

no more random elements, which led to the study

of stochastic games, which allowed the possibility of

random events occurring between decisions taken in

each period of time (Neyman and Sorin, 2003). To

do this, the theory reverted to studying simultaneous

games. Therefore, the situation is modelled as a re-

peated simultaneous game, where at every turn, the

game moves onto a different stage depending on a dis-

tribution that observes the action of every player and

the current stage. At every period all players receive

a utility, and at the end, the utility of each player is

calculated as the discounted sum.

In recent years, this approach has been expanded,

for example, by studying games where for every time

period there is a generation of players, who play a

noncooperative game, but in each successive time pe-

riod, there is a descendant of exactly one member of

the previous generation, and all these relatives have

to play cooperatively at the same time (Balbus and

Nowak, 2008), (Balbus et al., 2013), (Nowak, 2010),

(Wo´zny and Growiec, 2012). Another line of study

has been the modelling of altruism as a means of ob-

Becerril-Borja, R. and Montes-de-Oca, R.

Sequential Games with Finite Horizon and Turn Selection Process - Finite Strategy Sets Case.

DOI: 10.5220/0005696400440050

In Proceedings of 5th the International Conference on Operations Research and Enterprise Systems (ICORES 2016), pages 44-50

ISBN: 978-989-758-171-7

taining utility in a different way, by taking into ac-

count that the decision also inﬂuences the future, or

understanding altruism as something that beneﬁts the

future generations that play the game (Nowak, 2006)

(Saez-Marti and Weibull, 2005). Besides, some ad-

vances made recently have to do with the change of

stage and its effects on the other characteristics of the

game, for example, by restricting the options available

to each player through the use of penal codes, that is,

by making players themselves punish each other ac-

cording to the deviations incurred by some of them

(Kitti, 2011). All of these models are within the con-

text of repeated simultaneous games, and they com-

pute the utilities as a discounted sum of winnings ob-

tained in each repetition of the game.

The work presented here is an effort to model sit-

uations in a sequential frame, where individuals take

decisions in order, but where the order in which such

decisions are made is not given from the beginning,

but rather decided at every period of the game. The

decision of whose turn it is is made by a selection

process which, in the eyes of every player, is random,

so players can assign a probability distribution to the

selection process. Individuals only receive a utility at

the end of the game, depending on all the decisions

taken by all the players. Given this model, the exis-

tence of an equilibrium will be proven, and the result

will be extended to some other models: the ﬁrst one

allows changing the way each player models the se-

lection process as he learns new information; the sec-

ond one makes the turn selection process to be con-

ditioned on the decisions made by the players at all

previous turns; and a third one, where the strategy

sets are conditioned on the decisions made in previ-

ous turns. As far as we know, the models and results

presented here are a novel contribution to the theory

of sequential games.

The structure of this note is as follows: section

2 deﬁnes several models in the class of sequential

games with the turn selection process. The models

in subsections 2.1 and 2.2 consider a ﬁxed number

of decisions made by each player during the game. In

subsection 2.3 the base model is described and reﬁned

to obtain the models described in subsections 2.4–2.6.

In section 3, using the model of subsection 2.3 as

a base, the results that guarantee the existence of Nash

equilibria in the models are proved. The propositions

and their proofs can be easily modiﬁed to adapt them

for the other models presented before.

Finally, in section 4 an application is presented in

a “draft” of athletes for a certain sports league, where

the order of selection is not presented beforehand and

it changes throughoutthe game, as decisions are being

made.

2 SEQUENTIAL GAMES AND

TURN SELECTION PROCESS

2.1 One Turn per Player

Several concepts used in this article are standard in

game theory and can be consulted in a wide selec-

tion of books (Tadelis, 2013). Subscripts in elements

denote the player attributed to each element, super-

scripts denote the period of time considered, and left-

side subscripts are used for indexing sequences. A

glossary of terms can be found in the appendix at the

end of the article.

The main components of this model are:

• A set of players N = {1,2,... ,N}, with N ∈ N

ﬁxed.

• For each player j, S

is a ﬁnite set of pure strate-

gies, which contains all the possible decisions that

can be made throughout the game.

• A ﬁxed T ∈ N, which is the number of turns to be

played or the horizon of the game.

• A utility function u

: Σ → R for each player j,

where Σ is the cartesian product of S with itself T

times, where S = ∪

j=1

• A probability density p

: N → [0,1] of the distri-

bution that models the turn selection process ac-

cording to each player j.

Furthermore, M

will be deﬁned as the set of

mixed strategies for every player j obtained from the

set of pure strategies S

The ﬁrst thing to observe in any of the following

models is that, since no player has knowledge of the

turns at which they’ll be deciding, a strategy plan for

each of them must have an action for every turn, and

on each turn ℓ the decision should be conditioned on

the scenario s = (s

ℓ−1

,.. .,s

) made of the decisions

taken in periods 1 through ℓ − 1. Therefore, every

player j decides on s

a plan of conditioned strate-

gies for each period ℓ and for each possible scenario

of previously taken decisions. Now, the set of plans is

denoted by C

where only pure conditioned strategies

are used when confronted by any previous decisions

and the set of plans made of mixed conditioned strate-

gies by D

for each player j.

Finally, it is possible to build vectors of N compo-

nents, each of which is the strategy of a player. These

vectors are the proﬁles of conditioned strategies for

the game, whose sets will be denoted by C if they con-

sist completely of pure conditioned strategies, or D if

they allow the use of mixed conditioned strategies.

Sequential Games with Finite Horizon and Turn Selection Process - Finite Strategy Sets Case

In order to ﬁnd a solution to this model, a way

to evaluate each possible proﬁle is needed, while al-

lowing randomization in the turn selection process.

Therefore, an expectation operator is deﬁned for each

player j, and each proﬁle x ∈ D, by building the prod-

uct measure of the probability distribution of the turn

selection process and the mixed strategy that is being

evaluated.

In this model, exactly one decision will be allowed

per player, even though, the period of time in which

that decision will be made is not known beforehand.

This way, the expected utility of each player j for a

given x = (x

,.. .,x

) ∈ D will be deﬁned as

(x) =

∑

,...,n

)∈P (N )

∑

∈S

···

∑

∈S

,.. .,s

)

× x

| s

N−1

,.. .,s

)···x

where P (N ) is the set of permutations of the set N .

Based on this, a Nash equilibrium can be deﬁned

as a strategy proﬁle x

∗

∈ D such that, for every player

j ∈ N ,

∗

) ≥ E

∗

− j

)

for all x

∈ D

, where x

− j

is the partial proﬁle that

considers the strategies of all players but j. This way,

a proﬁle (y

− j

) means that the strategies of y are

considered for every player but j, for whom the strat-

egy given by z

is used. This deﬁnition of Nash equi-

librium will be used through all the models presented,

where the respective expected utility function is con-

sidered.

2.2 Fixed Number of Turns per Player

Next the model will be generalized by allowing each

player to make more than one decision. The number

of decisions made, however, is ﬁxed beforehand for

each player, but as before, the periods in which these

decisions are made are not known a priori.

If the number of decisions made in each period for

each player is given by the vector m = (m

,.. .,m

then the expected utility for player j is deﬁned as:

(x) =

∑

,...,n

)∈P

(N )

∑

∈S

···

∑

∈S

,.. .,s

)

× x

| s

T−1

,.. .,s

)···x

)

where P

(N ) is the set of permutations of the set N

with m

repetitions of k, and T =

∑

j∈N

2.3 Unknown Number of Turns per

Player: Base Model

Now, the attention is shifted to the central model. In-

stead of having information of how many decisions a

player will be making, that information will be hidden

and all players will be allowed to potentially make as

many decisions as possible in T periods of time. In

this case, the expected utility of player j will be de-

ﬁned when facing the proﬁle x ∈ D as:

(x) =

∑

∈N

∑

∈S

···

∑

∈N

∑

∈S

,.. .,s

)

× x

| s

T−1

,.. .,s

)···x

2.4 Updated Models of the Turn

Selection Process in each Period

In the base model of subsection 2.3, each player was

thought to be making his predictions of the behaviour

of the turn selection process from the beginning of

the game and not changing thereafter. However, this

a priori distribution may not be accurate throughout

the game, or the player may learn new information of

its behaviour by observing how players are selected

at each turn. Therefore, a player has to be allowed to

have different distributions for each period.

Instead of having a ﬁxed probability density p

each player has a vector (p

, p

,.. ., p

) of probabil-

ity densities for each period. The expected utility for

player j will be deﬁned as

(x) =

∑

∈N

∑

∈S

···

∑

∈N

∑

∈S

,.. .s

)

× x

| s

T−1

,.. .,s

)···x

2.5 Conditioned Turn Selections on

Previous Decisions

In the base model, only the mixed strategies are de-

pendent on the decisions made in the previous turns.

To approach the usual modelling of games by stages

every element should be allowed to change according

to the decisions made previously. Though originally

it was expected to be conditioned as a Markov-like

structure, that is, the structure would depend only on

the immediate previous decision made in the game, it

is possible to generalize it, conditioning the turn se-

lection process on every single decision taken so far.

This also takes into account conditioning only the way

the process itself has behaved,that is, changingthe se-

lection according to which players have been picked

to make decisions through the game, and not their de-

cisions.

In this case, the expected utility function for each

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

player j is deﬁned in the following way:

(x) =

∑

∈N

∑

i∈S

···

∑

∈N

∑

∈S

u(s

,.. .,s

)

× x

| s

T−1

,.. .s

| s

T−1

,.. .,s

)···

× x

2.6 Strategy Sets Changing According

to Period

To approach the modelling per stages from a differ-

ent perspective, the sets of strategies will be able to

change according to the period of time in which the

decision is made. This also can be modiﬁed to take

into account the fact that strategy sets may change ac-

cording to the previously made decisions.

For this model, the expected utility function for

each player j is deﬁned as follows:

(x) =

∑

∈N

∑

∈S

···

∑

∈N

∑

∈S

u(s

,.. .,s

)

× x

| s

T−1

,.. .s

| s

T−1

,.. .,s

)···

× x

)

or, accordingly, the strategy sets are conditioned by

the previously made decisions, that is:

(x) =

∑

∈N

∑

∈S

···

∑

∈N

∑

∈S

T−1

,...,s

)

u(s

,.. .,s

)

× x

| s

T−1

,.. .s

| s

T−1

,.. .,s

)···

× x

3 EXISTENCE OF NASH

EQUILIBRIA IN THE MODELS

In this section a series of results is shown that ensures

the existence of Nash equilibria in the base model of

subsection 2.3. These results can also be adapted for

each of the other models proposed, to prove the exis-

tence of equilibria in them as well.

To do this, it is necessary to prove the sufﬁcient

conditions of Kakutani’s ﬁxed-point theorem (Zei-

dler, 1986) for the best response correspondence as-

sociated with the expected utility function deﬁned for

each model. The proofs are similar in all the models,

in some cases only needing to rewrite the elements in

the expected utility function, which don’t affect the

proofs themselves.

Theorem 1. The expected utility function is a con-

tinuous function in each player’s plan of conditioned

strategies.

Proof. Observe that after expanding the sums in the

expected utility function, each component is constant,

linear, quadratic, cubic, etc., up to a T-ic function ac-

cording to the periods in which player j chooses, with

the rest of the components scaling the function. As

such, each component is a continuous function, and

the sum of all of them is also continuous.

Theorem 2. The set D is a non-empty, compact and

convex subset of R

for a suitable q.

Proof. For each scenario s = (s

ℓ−1

,.. .,s

), the mixed

strategy of player j given s lies in a a

ℓ

-simplex, where

ℓ

is the number of strategies available to player j at

period ℓ. For each player j, the set of plans is the

cartesian product of these simplices for all the possi-

ble scenarios, and the set D is the cartesian product

of the sets of plans for every player. As a cartesian

product of simplices, it is a non-empty, compact and

convex subset of R

Deﬁne for each player j, BR

− j

) his best re-

sponse correspondence for the partial proﬁle x

− j

which maps x

− j

to the set of plans of mixed con-

ditioned strategies x

′

∈ D

such that E

− j

′

) ≥

− j

) for every y

∈ D

Theorem 3. The best response correspondence

BR: D → D, given by

BR(x

∗

) = (BR

∗

−1

),BR

∗

−2

),...,BR

∗

−N

)),

is a non-empty correspondence with a closed graph.

Proof. Since the expected utility function is a con-

tinuous function deﬁned on a compact set, it must

achieve its maximum at some point ˆx

∈ D

for each

− j

∈ D

− j

for every player j. Therefore, BR

is non-

empty for every player j and every x ∈ D, which im-

plies that BR(x) is a non-empty correspondence for

every x ∈ D.

Now, consider (

∞

h=1

as a sequence of strategy

proﬁles and (

′

)

∞

h=1

as the associated sequence of

the best responses, that is,

′

∈ BR(

x). Let x

∗

lim

h→∞

x and x

′∗

= lim

h→∞

′

. Fix player j, then

′

∈ BR

(

− j

), which means that

(

− j

′

) ≥ E

(

− j

for any y

∈ D

. As the expected utility function is

continuous in each player’s plan of strategies, it is

possible to take limits on both sides while preserving

the inequality, which means that

lim

h→∞

(

− j

′

) ≥ lim

h→∞

(

− j

)

and, interchanging the order of limits and sums,

∗

− j

′∗

) ≥ E

∗

− j

)

for any y

∈ D

. This implies that x

′∗

∈ BR(x

∗

− j

) for

each player j, and, therefore, x

′∗

∈ BR(x

∗

Sequential Games with Finite Horizon and Turn Selection Process - Finite Strategy Sets Case

Finally, the convexity of the best response corre-

spondence will require a few more arguments. Given

a plan of strategies x

, a similar plan of strategies y

for scenario s = (s

ℓ−1

,.. .,s

) is deﬁned as the plan

in which all mixed strategies are the same in x

and

, except for the one conditioned by s, which is re-

placed by a pure conditioned strategy s

ℓ

∈ supp(x

,s),

where supp(x

,s) = {s

ℓ

∈ S

| x

ℓ

| s) > 0} is the

support of the conditioned strategy x

for scenario

s. SM

| s) denotes the set of plans of condi-

tioned strategies similar to x

of player j for scenario

s = (s

ℓ−1

,.. .,s

With this in mind, it is possible to start our string

of results.

Lemma 1. Let x ∈ D be such that for player j, x

∈

− j

). Then, for any scenario s = (s

ℓ−1

,.. .,s

and any two plans y

∈ SM

| s):

− j

) = E

− j

Proof. Assume that E

− j

) > E

− j

) for

some y

∈ SM(x

| s). Observe that E

can be par-

titioned in two components of the sums: those where

the scenario s occurs in the ﬁrst ℓ − 1 periods and

the player j is chosen to make the decision in pe-

riod ℓ, and those where one of the previous conditions

fails. The latter ones are not affected by replacing y

with z

, so the focus will be only on the former ones,

where the inequality still holds. If this is the case,

then in x

the probability assigned to z

could be re-

duced to zero, and the probability of y

could be in-

creased to x

) + x

), which would increase the

expected utility obtained. But this is a contradiction

to the fact that x

was a best response to x

− j

. There-

fore, the strict inequality cannot hold, and so equality

is obtained.

Lemma 2. Let x ∈ D be such that for player j, x

∈

− j

). Then, for any scenario s = (s

ℓ−1

,.. .,s

and any plan y

∈ SM

| s),

(x) = E

− j

Proof. As in the previous lemma, it is possible to

partition the sum in two. The relevant part of the

sums of E

(x) can be written as the weighted sum

of the corresponding relevant parts of E

− j

) for

all z

∈ SM

| s), where the weights are the prob-

abilities assigned to each strategy in x

(· | s). But

by lemma 1, all these parts have the same value, so

it is possible to replace them by the relevant part

of E

− j

). Being a convex combination of the

same value, it amounts to exactly the relevant part of

− j

), proving the equality.

The following corollary is obtained as a conse-

quence of the previous lemma.

Corollary 1. Let x ∈ D be such that for player j, x

∈

− j

). Then, for any scenario s = (s

ℓ−1

,.. .,s

the plan of strategies y

obtained by substituting the

conditioned strategy in scenario s for any mixed strat-

egy which has a support that is a subset of the support

of x

, satisﬁes

(x) = E

− j

Observe that if exactly two mixed strategies were

changed by pure strategies, the equality would be ob-

tained by applying lemma 1 twice, changing one by

one each strategy. And it is possible to change two

mixed strategies for any other two mixed strategies

as well, and guarantee that the equality still holds,

by applying lemma 2 to change the strategies one

by one. It is possible to extend this process to any

number of changes in the strategies. Given the pre-

viously described process, the following result is ob-

tained which generalizes the previous lemmas.

Theorem 4. Let x ∈ D be such that for player j, x

∈

− j

). Then, for any scenario s = (s

ℓ−1

,.. .,s

and any two plans of strategies y

∈ D

with their

support a subset of the support of x

, we have:

− j

) = E

− j

)

Using the previous result, we can easily prove the

last condition needed for the best response correspon-

dence.

Theorem 5. The best response correspondence BR is

convex.

Proof. Let x ∈ D, ﬁx j ∈ N , and consider x

′

′′

∈

− j

). Observe that any convex combination of

′

and x

′′

is a plan of strategies in D

. Since the con-

vex combination of x

′

and x

′′

is a plan of strategies

with its support, which is a subset of the support of

plans, representing the best response, by theorem 4,

the expected utility of the convex combination is the

same as the expected utility for any best response.

Therefore, the convex combination is also a best re-

sponse.

All the previous results give the conditions of

Kakutani’s ﬁxed-point theorem for the best-response

correspondence, which guarantees the existence of a

ﬁxed point; this ﬁxed point is our Nash equilibrium.

Therefore, the central theorem for the model is ob-

tained.

Theorem 6. Every sequential game with ﬁnite hori-

zon and turn selection process and ﬁnite strategies

sets has at least one Nash equilibrium.

ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems

4 AN APPLICATION OF THE

MODEL: PICKING

TEAMMATES

In the following example an application of some of

the models analysed in the previous sections is shown.

To be precise, the example is a combination of two

models, where the probabilities of the turn selection

process, as well as the strategy sets change for each

period conditioned on the decisions made before.

In a certain sports league, everyyear the two teams

participating have to choose between certain college

athletes to become part of their team. This year they

have to pick between athlete A and athlete B. Both

of them know that A, becoming a teammate, gives a

utility of 1 whereas B gives a utility of 2. The utility

obtained by each team after the picking is the sum of

the utilities of the athletes they have picked.

However, the process involved goes like this: in

each of the two picking periods, each team has a prob-

ability of being selected to make a choice. Since team

1 won last season, they get a probability of being se-

lected in the ﬁrst period of 1/3, whereas team 2 gets

a probability of 2/3. If athlete A is selected ﬁrst, the

team that picked him has its probability of being se-

lected for the second period reduced to half (which

goes to the other team). If athlete B is selected ﬁrst,

the reduction in probability is to a third of the original

probability. Then, with the new probabilities, a team

is selected for the second pick, which automatically

chooses the athlete not picked in the ﬁrst round.

To solve the problem, ﬁrst the probabilities in each

instance are calculated. For the strategies sets, for

team 1, there is S

= {A

} and for team 2, S

} (the subscripts are to identify who made the

pick in the ﬁrst round when conditioning). Since both

players know the probabilities in the picking process

they model it the same way, so the subscript of p

removed. This way, there are

p(1) = 1/3 p(2) = 2/3

p(1 | A

) = 1/6 p(2 | A

) = 5/6

p(1 | B

) = 1/9 p(2 | B

) = 8/9

p(1 | A

) = 2/3 p(2 | A

) = 1/3

p(1 | B

) = 7/9 p(2 | B

) = 2/9

Now, the expected utility function is calculated

for each player, inputting the probabilities for each

possible scenario, and the utilities received in each

case. Observe that since the second pick is automatic,

| B

), x

| B

), x

| A

), x

| A

) are

all equal to one, for j ∈ {1, 2}, and the other mixed

strategies are equal to zero. The expected utilities ob-

tained are:

(x) =

)

10x

)

and

(x) =

)

−

10x

)

To maximize their expected utility functions,

) = 1 and x

) = 0, which means that in

case team 1 is chosen for the ﬁrst round, they should

choose athlete A, but if team 2 is chosen ﬁrst, they

should choose athlete B.

5 CONCLUSIONS

This work introduces a model for situations where the

individuals may not know the moment in which they

make a decision, or its placement according to the de-

cisions of other players. It also introduces a model

to create situations which allow a randomization pro-

cess to equalize players, whether it is by being able

to make decisions earlier, or by receiving information

that is useful in later periods. It is also a good model

for buying and selling scarce goods, or even goods

of different utility value. The models introduced here

are sequential, which is a more natural way of viewing

situations that happen in real life, and they comprise a

wide array of possibilities, by allowing various parts

of the model to be changing throughout time, or even

combining several of these changes at the same time.

However, the example presented has the basic el-

ements to illustrate the model, but it is not made with

larger sets of strategies, more players, or more periods

of time, because, to solve the model, the calculations

made become unbearable very quickly, and prone to

errors if made by hand, which then requires a com-

puter program to be created in order to analyze each

situation. And this becomes the only method as the

problem grows, since we start ﬁnding crossed prod-

ucts of different strategies, which become increas-

ingly hard to solve.

For further work, the extension to inﬁnite sets of

strategies follows naturally. Also, possible ramiﬁca-

tions include studying a way to change the problem

into an optimization problem to ﬁnd Nash equilib-

ria, allowing an external source of noise to intervene

in the decision process and to study the convergence

of sequences of approximate solutions within each of

these models.

Sequential Games with Finite Horizon and Turn Selection Process - Finite Strategy Sets Case

ACKNOWLEDGEMENTS

This work was partially supported by CONACYT

(Mexico) and ASCR (Czech Republic) under Grants

No. 171396 and 283640.

The authors would like to dedicate this work to

Dr. Gabriel Zacar´ıas Espinoza, a dear and kind friend.

Also the ﬁrst author would like to dedicate this work

to his grandmother Matilde Fonseca S´anchez who al-

ways supported him, as well as to Dr. Rafael Ed-

mundo Morones Escobar, a great professor who was

always kind and a good friend. All of you will be

missed dearly.

Finally, we’d like to thank our families and friends

for the support throughout this research, and the sup-

port we know you’ll give us as we continue in this

journey. Thank you for always being there.

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APPENDIX

• N is the number of players in the game.

• N is the set of players in the game.

• T is the horizon of the game, that is, the number

of turns to be played in the game.

• S

is the set of pure strategies for player j, which

is considered to be ﬁnite.

• S = ∪

j=1

• M

is the set of mixed strategies for player j, de-

ﬁned as the set of probability distributions that can

be assigned to the set of pure strategies of player

• M = ∪

j=1

• x

− j

is the strategy for all players except j obtained

from a mixed strategy x by deleting the strategy

played by j.

• s = (s

ℓ−1

,.. .,s

) is a scenario for period ℓ ∈

{1,· ·· ,T}, which is a vector consisting of the

decisions s

,.. .s

ℓ−1

that were made in periods 1

through ℓ − 1.

• C

is the set of plans made of conditioned pure

strategies for player j. That is, for every possi-

ble scenario, the player makes his decision with a

pure strategy, conditioned on the scenario that he

is facing.

• D

is the set of plans made of conditioned mixed

strategies for player j, that is, the player assigns a

mixed strategy to each possible scenario that can

be faced in the game.

• C = ∪

j=1

• D = ∪

j=1

• P (N ): the set of permutations of N .

• P

(N ) is the set of permutations of the set made

of m

repetitions of k for each k ∈ N , if m =

,.. .,m

• BR

− j

) is the best response correspondence of

player j for the partial proﬁle x

− j

, which maps x

− j

to the set of plans of mixed conditioned strategies

′

∈ D

such that E

− j

′

) ≥ E

− j

) for ev-

ery y

∈ D

• BR(x) = (BR

−1

),BR

−2

),...,BR

−N

)) is

the best response correspondence for the mixed

strategy x ∈ D.

• SM

| s) is the set of plans of conditioned

strategies similar to x

of player j for scenario s.

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