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-
44
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
Copyright
c
2016 by SCITEPRESS Science and Technology Publications, Lda. All rights reserved
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
j
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
j
: Σ R for each player j,
where Σ is the cartesian product of S with itself T
times, where S =
N
j=1
S
j
.
A probability density p
j
: N [0,1] of the distri-
bution that models the turn selection process ac-
cording to each player j.
Furthermore, M
j
will be defined as the set of
mixed strategies for every player j obtained from the
set of pure strategies S
j
.
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 the decision should be conditioned on
the scenario s = (s
1
,.. .,s
1
) made of the decisions
taken in periods 1 through 1. Therefore, every
player j decides on s
j
a plan of conditioned strate-
gies for each period and for each possible scenario
of previously taken decisions. Now, the set of plans is
denoted by C
j
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
j
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
45
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
1
,x
2
,.. .,x
N
) D will be defined as
E
j
(x) =
(n
1
,...,n
N
)P (N )
s
1
S
n
1
···
s
N
S
n
N
u
j
(s
N
,.. .,s
1
)
× x
n
N
(s
N
| s
N1
,.. .,s
1
)p
j
(n
N
)···x
n
1
(s
1
)p
j
(n
1
),
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
(x
) E
j
(x
j
,x
j
)
for all x
j
D
j
, where x
j
is the partial profile that
considers the strategies of all players but j. This way,
a profile (y
j
,z
j
) means that the strategies of y are
considered for every player but j, for whom the strat-
egy given by z
j
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
1
,.. .,m
N
),
then the expected utility for player j is defined as:
E
j
(x) =
(n
1
,...,n
T
)P
m
(N )
s
1
S
n
1
···
s
T
S
n
T
u
j
(s
T
,.. .,s
1
)
× x
n
T
(s
T
| s
T1
,.. .,s
1
)p
j
(n
T
)···x
n
1
(s
1
)p
j
(n
1
)
where P
m
(N ) is the set of permutations of the set N
with m
k
repetitions of k, and T =
jN
m
j
.
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:
E
j
(x) =
n
1
N
s
1
S
n
1
···
n
T
N
s
T
S
n
T
u
j
(s
T
,.. .,s
1
)
× x
n
T
(s
T
| s
T1
,.. .,s
1
)p
j
(n
T
)···x
n
1
(s
1
)p
j
(n
1
).
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
j
,
each player has a vector (p
1
j
, p
2
j
,.. ., p
T
j
) of probabil-
ity densities for each period. The expected utility for
player j will be defined as
E
j
(x) =
n
1
N
s
1
S
n
1
···
n
T
N
s
T
S
n
T
u
j
(s
T
,.. .s
1
)
× x
n
T
(s
T
| s
T1
,.. .,s
1
)p
T
j
(n
T
)···x
n
1
(s
1
)p
1
j
(n
1
).
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
46
player j is defined in the following way:
E
j
(x) =
n
1
N
s
1
iS
n
1
···
n
T
N
s
T
S
n
T
u(s
T
,.. .,s
1
)
× x
n
T
(s
T
| s
T1
,.. .s
1
)p
j
(n
T
| s
T1
,.. .,s
1
)···
× x
n
1
(s
1
)p
j
(n
1
).
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:
E
j
(x) =
n
1
N
s
1
S
1
n
1
···
n
T
N
s
T
S
T
n
T
u(s
T
,.. .,s
1
)
× x
n
T
(s
T
| s
T1
,.. .s
1
)p
j
(n
T
| s
T1
,.. .,s
1
)···
× x
n
1
(s
1
)p
j
(n
1
)
or, accordingly, the strategy sets are conditioned by
the previously made decisions, that is:
E
j
(x) =
n
1
N
s
1
S
1
n
1
···
n
T
N
s
T
S
n
T
(s
T1
,...,s
1
)
u(s
T
,.. .,s
1
)
× x
n
T
(s
T
| s
T1
,.. .s
1
)p
j
(n
T
| s
T1
,.. .,s
1
)···
× x
n
1
(s
1
)p
j
(n
1
).
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
q
for a suitable q.
Proof. For each scenario s = (s
1
,.. .,s
1
), the mixed
strategy of player j given s lies in a a
i
-simplex, where
a
i
is the number of strategies available to player j at
period . 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
q
.
Define for each player j, BR
j
(x
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
j
D
j
such that E
j
(x
j
,x
j
)
E
j
(x
j
,y
j
) for every y
j
D
j
.
Theorem 3. The best response correspondence
BR: D D, given by
BR(x
) = (BR
1
(x
1
),BR
2
(x
2
),...,BR
N
(x
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
j
D
j
for each
x
j
D
j
for every player j. Therefore, BR
j
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
x)
h=1
as a sequence of strategy
profiles and (
h
x
)
h=1
as the associated sequence of
the best responses, that is,
h
x
BR(
h
x). Let x
=
lim
h
h
x and x
′∗
= lim
h
h
x
. Fix player j, then
h
x
j
BR
j
(
h
x
j
), which means that
E
j
(
h
x
j
,
h
x
j
) E
j
(
h
x
j
,y
j
),
for any y
j
D
j
. 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
E
j
(
h
x
j
,
h
x
j
) lim
h
E
j
(
h
x
j
,y
j
)
and, interchanging the order of limits and sums,
E
j
(x
j
,x
′∗
j
) E
j
(x
j
,y
j
)
for any y
j
D
j
. This implies that x
′∗
j
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
47
Finally, the convexity of the best response corre-
spondence will require a few more arguments. Given
a plan of strategies x
j
, a similar plan of strategies y
j
for scenario s = (s
1
,.. .,s
1
) is defined as the plan
in which all mixed strategies are the same in x
j
and
y
j
, except for the one conditioned by s, which is re-
placed by a pure conditioned strategy s
j
supp(x
j
,s),
where supp(x
j
,s) = {s
j
S
j
| x
j
(s
j
| s) > 0} is the
support of the conditioned strategy x
j
for scenario
s. SM
j
(x
j
| s) denotes the set of plans of condi-
tioned strategies similar to x
j
of player j for scenario
s = (s
1
,.. .,s
1
).
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
BR
j
(x
j
). Then, for any scenario s = (s
1
,.. .,s
1
),
and any two plans y
j
,z
j
SM
j
(x
j
| s):
E
j
(x
j
,y
j
) = E
j
(x
j
,z
j
).
Proof. Assume that E
j
(x
j
,y
j
) > E
j
(x
j
,z
j
) for
some y
j
,z
j
SM(x
j
| s). Observe that E
j
can be par-
titioned in two components of the sums: those where
the scenario s occurs in the first 1 periods and
the player j is chosen to make the decision in pe-
riod , and those where one of the previous conditions
fails. The latter ones are not affected by replacing y
j
with z
j
, so the focus will be only on the former ones,
where the inequality still holds. If this is the case,
then in x
j
the probability assigned to z
j
could be re-
duced to zero, and the probability of y
j
could be in-
creased to x
j
(y
j
) + x
j
(z
j
), which would increase the
expected utility obtained. But this is a contradiction
to the fact that x
j
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
BR
j
(x
j
). Then, for any scenario s = (s
1
,.. .,s
1
),
and any plan y
j
SM
j
(x
j
| s),
E
j
(x) = E
j
(x
j
,y
j
).
Proof. As in the previous lemma, it is possible to
partition the sum in two. The relevant part of the
sums of E
j
(x) can be written as the weighted sum
of the corresponding relevant parts of E
j
(x
j
,z
j
) for
all z
j
SM
j
(x
j
| s), where the weights are the prob-
abilities assigned to each strategy in x
j
(· | 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
(x
j
,y
j
). Being a convex combination of the
same value, it amounts to exactly the relevant part of
E
j
(x
j
,y
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
BR
j
(x
j
). Then, for any scenario s = (s
1
,.. .,s
1
),
the plan of strategies y
j
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
j
, satisfies
E
j
(x) = E
j
(x
j
,y
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
BR
j
(x
j
). Then, for any scenario s = (s
1
,.. .,s
1
),
and any two plans of strategies y
j
,z
j
D
j
with their
support a subset of the support of x
j
, we have:
E
j
(x
j
,y
j
) = E
j
(x
j
,z
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
,x
′′
j
BR
j
(x
j
). Observe that any convex combination of
x
and x
′′
is a plan of strategies in D
j
. 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
48
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
1
= {A
1
,B
1
} and for team 2, S
2
=
{A
2
,B
2
} (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
j
is
removed. This way, there are
p(1) = 1/3 p(2) = 2/3
p(1 | A
1
) = 1/6 p(2 | A
1
) = 5/6
p(1 | B
1
) = 1/9 p(2 | B
1
) = 8/9
p(1 | A
2
) = 2/3 p(2 | A
2
) = 1/3
p(1 | B
2
) = 7/9 p(2 | B
2
) = 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,
x
j
(A
j
| B
1
), x
j
(A
j
| B
2
), x
j
(B
j
| A
1
), x
j
(B
j
| A
2
) are
all equal to one, for j {1, 2}, and the other mixed
strategies are equal to zero. The expected utilities ob-
tained are:
E
1
(x) =
25
27
+
x
1
(A
1
)
27
+
10x
2
(A
2
)
27
and
E
2
(x) =
16
9
+
7x
1
(A
1
)
27
10x
2
(A
2
)
27
.
To maximize their expected utility functions,
x
1
(A
1
) = 1 and x
2
(A
2
) = 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
49
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.
REFERENCES
Balbus, Ł. and Nowak, A. S. (2008). Existence of per-
fect equilibria in a class of multigenerational stochas-
tic games of capital accumulation. Automatica,
44(6):1471–1479.
Balbus, Ł., Reffett, K., and Wo´zny, Ł. (2013). A con-
structive geometrical approach to the uniqueness of
Markov stationary equilibrium in stochastic games of
intergenerational altruism. Journal of Economic Dy-
namics and Control, 37(5):1019–1039.
Fudenberg, D. and Tirole, J. (1991). Game Theory. The
MIT Press.
Kitti, M. (2011). Conditionally stationary equilibria in dis-
counted dynamic games. Dynamic Games and Appli-
cations, 1(4):514–533.
Neyman, A. and Sorin, S., editors (2003). Stochastic Games
and Applications. Springer.
Nowak, A. S. (2006). On perfect equilibria in stochas-
tic models of growth with intergenerational altruism.
Economic Theory, 28(1):73–83.
Nowak, A. S. (2010). On a noncooperative stochastic game
played by internally cooperating generations. Journal
of Optimization Theory and Applications, 144(1):88–
106.
Saez-Marti, M. and Weibull, J. W. (2005). Discounting and
altruism to future decision-makers. Journal of Eco-
nomic Theory, 122(2):254–266.
Tadelis, S. (2013). Game Theory. Princeton University
Press.
Wo´zny, Ł. and Growiec, J. (2012). Intergenerational inter-
actions in human capital accumulation. The BE Jour-
nal of Theoretical Economics, 12(1).
Zeidler, E. (1986). Nonlinear Functional Analysis and its
Applications I: Fixed-Point Theorems. Springer.
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
j
is the set of pure strategies for player j, which
is considered to be finite.
S =
N
j=1
S
j
.
M
j
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
j.
M =
N
j=1
M
j
.
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
1
,.. .,s
1
) is a scenario for period
{1,· ·· ,T}, which is a vector consisting of the
decisions s
1
,.. .s
1
that were made in periods 1
through 1.
C
j
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
j
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 =
N
j=1
C
j
.
D =
N
j=1
D
j
.
P (N ): the set of permutations of N .
P
m
(N ) is the set of permutations of the set made
of m
k
repetitions of k for each k N , if m =
(m
1
,.. .,m
N
).
BR
j
(x
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
x
j
D
j
such that E
j
(x
j
,x
j
) E
j
(x
j
,y
j
) for ev-
ery y
j
D
j
.
BR(x) = (BR
1
(x
1
),BR
2
(x
2
),...,BR
N
(x
N
)) is
the best response correspondence for the mixed
strategy x D.
SM
j
(x
j
| s) is the set of plans of conditioned
strategies similar to x
j
of player j for scenario s.
ICORES 2016 - 5th International Conference on Operations Research and Enterprise Systems
50