1

Survival strategies in a dystopic economy using a

player–versus–environment dynamic stochastic game1

Edgardo Manuel Jopson

De La Salle University

School of Economics

Abstract: In a post-apocalyptic dystopia, the struggle to survive is more pronounced than the usual.

From the risk of death ever present given the level of violence and cruel environment caused by

catastrophic events, an individual experiences a player–versus–environment (PvE) game that gambles its

life in order to gather resources for survival. This paper attempts to express the player’s possible

strategies in order to survive these conditions, which would largely depend on risk propensities and

survival. We propose that the player’s survival would depend on his or her demand for resources,

governed by an Epstein-Zin Utility function (1990), which incorporates risk and consumption

preferences and the players adaptability is accounted for by a dynamic Cobb-Douglas production

function which takes into consideration the ability and survival skills via a multiplier which changes

after the second stage of the game in order to account for the player’s learning curve.

JEL CLASSIFICATION: C73, C63, C60, D81, D82

Keywords: game theory; chaos theory; dystopia; player–versus-environment survival analysis; dynamic

stochastic games

1 The proponent wishes to thank the faculty of the School of Economics with their helpful comments from the Brown Bag Session held on 27

January 2016 at De La Salle University as well as the expertise and insight of Dr. Kristine Joy Carpio of the Mathematics Department of De La

Salle University, most especially on the mathematical propositions and proofs. The introduction and the subsection Survival and Adaptation of

chapter 2 have been presented at the De La Salle University Research Congress held on 9 March 2016 with Dr. Nelson Arboleda as Moderator.

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1. Introduction

The pursuit of economic stability has always been in the mindset of humanity; that is, we

are more inclined to prefer average outcomes whatever the state–of–the–world may be, over

extremes. Humanity also prefers to have stable relationships amongst each other and devotes a

great deal of resources to do so; institutions such as the United Nations, the World Bank, the World

Health Organization and the World Trade Organization, which ensures that humanity does not

stray to the path of self-destruction, by insuring sustainable development, combating terrorism,

promoting gender equality, securing food production. Overall, these institutions are in place in

order to maintain peace and security amongst member nations (United Nations, 2016).

However, apart from terrorism, biological epidemics and natural disasters in the twenty-

first century alone, the number of armed assaults has more than doubled from ten years ago, as

well as for bombings and explosive terrorism. Not only have their incidences doubled, but their

rate of success as well. From the beginning of the twenty-first century up to 2014, terrorist attacks

have become 500% more successful.

Figure 1. Incidence of armed assaults, assassinations, bombings/explosions and successful terrorist attacks (Source: START,

Global Terrorism Database)

Aside from the threat of widespread terrorism and violence, some scholars are taking the

problem of uncontrolled, rapid development in technology as a legitimate threat to humanity’s

survival. In the situation of a technological fallout, the concept of the singularity and Moore’s Law

has been a major topic for academic debate. In short, Moore’s law states that the computational

power of transistors in a computer doubles every 18 months – which translates that computers

have an exponential growth in terms of intelligence, while humans do not. Futurists such as Ray

Kurzweil predict that humans will be dependent on machines in the middle of the twenty-first

century (Diamandis & Kotler [2012], Chalmers [2010]), the question of stopping these unregulated

computers to decide to take control would be raised. Despite this statement sounding rather in the

realm of science–fiction, computer scientists and physicists are considering this far from a myth,

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but rather an impending doom for humanity: in the likes of Stephen Hawking, Elon Musk, and

Vernor Vinge (Luckerson, 2014).

Given definite characteristics of the agents interacting in a post apocalypse we explain how

a typical player would behave given the chaotic environment where the risk changes over time at

random using game theory.

2. THEORETICAL FRAMEWORK

Survival and Adaptation

This paper attempts to model the behavior of the individual that endeavors to survive the

dystopic landscape set using game theory. In this model we assume that our representative player,

Player A, which we denote as X is mobile and makes decisions to consume resources (optimize

survival) and invest in capital (gather food and resources). The risk of death is denoted by and

the intertemporal elasticity of consumption denoted by some .2 Thus

11

1 (1 )( ) max ( *, , , ) [ ( ) ]

ZX

S z z c k E S z (1)

where

1( *, , , ),

1

z f c k

S(z) denotes the survivability function of the individual as he or she traverses the

dystopian post-apocalyptic world. which is a function of consumption *c at the optimal

consumption in terms of calorie intake, investment in capital resources k , the “will to survive”

denoted by , and investment of the individual in developing strength, agility and mental alertness

in order to effectively survive the dystopic landscape denoted by . Keeping in mind that the

game environment is a harsh dystopia; filled with other competitors which impose a risk to Player

A.

Similarly Player B, which we denote as Y also has a utility function similar to Player A

(assuming homogeneous players in the game)

11

1 (1 )( ) max ( * , , ) [ ( ) ]

ZY

S z z c k E S z (2)

2 denotes the Arrow-Pratt relative risk aversion coefficient and denotes the intertemporal elasticity of substitution

4

The player is responsible for allocating his labor hours and leisure depending on the risks

involved in production. Since we are situated in a dystopic economy (hence, a diseconomy3),

assume a constant threat of death while gathering resources, hence leisure does not exists. Rather,

the player will be sheltering herself, minimizing the risk of death; denoted as hide, while labor is

denoted as seek. When the player chooses to seek resources, she is able to accumulate resources

enough for her to live for another time period, called a stockpile. Given these assumptions, we

model a dynamic Cobb-Douglas production function:

1

1

( ) [ ( , ) ( , ) ]

t t t t

t

A q E L w r K w r (3)

In this adaptation function, we denote A(q) as our adaptability function where it is a

function of Labor and Capital, allocated within a Constant Elasticity of Substitution, a multiplier

t and t denoted for the abilities and an independent force of mortality, respectively.

[ , ] tw f p (4)

[ , ] tr f p (5)

1

1

( , ( ), )t

tt t n t t

t

f A q E

(6)

( )(1 )

1 ( )

t t

f tf

F t (7)

In equations 4 to 7, t denotes the action Seek for period t while

t denotes the action Hide

also for period t. Furthermore, we set wages and rent as a function of the action Seek which allows

our player to “purchase” output, and prices, p, which denotes the resource cost for gathering

resources. t is a function of a learning curve which is affected by the previous period’s

adaptability function multiplied by a learning curve multiplier discounted over time, and the game

environment denoted by tE . Since ability is an estimate, we take into account an error term, else

the player’s learning varied from the true estimate. Λt is the force of mortality that takes into

consideration the probability of death (Konstantopoulos, 2006). If at that specific period the player

dies, A(q) then approaches zero, terminating the game. A component of some function involving

the Hide option is made.

Note that Player A is now playing a game of survival against a new environment profile. The

environment profile Et for this study is not only limited within the forces of nature, but a collection

of natural hazards 1,...,( )k k me and optimizing agents 1,...,( )l l nb

such that

3 In the situation of a dystopia, we assume that the market is nonexistent, which goes without saying that law and order is also nonexistent.

If one can imagine the film Book of Eli (Hughes & Hughes, 2010), trade is done via a common commodity, such as water, or in the form of

barter. The important element that has to be considered in this type of economy is that there is no government that enforces the law. This

means that the players in this model are self-reliant and are responsible for their own survival.

5

( )kk e ke P d e (8)

( ), ( ),l l ll b b bb P f S z A q (9)

( )kP e denotes the probability of event ke and ( )kd e denotes the disutility caused by the

event, such that0 ( ) 1kP e and ( ) 0kd e . Furthermore, ( )lP b denotes the probability of Player A of

encountering lb and ( ), ( ),l lb bf S z A q

which is optimizing choice of lb which will affect the

chance of survival4 for Player A.

Without loss of generality, let m = 2 denoting two natural events, and n = 2 where there are

only two players, X and Y and both players have the same functional form for survival and

adaptation, and they are both experiencing the same environment. Illustrating their respective

survival strategy using a decision tree for two periods and n periods

Figure 2. Hide and Seek Decision Tree (2 stage) (von Auer, 1998)

Extending the game up to zn stages, we obtain

Figure 3. Hide and Seek Decision Tree (n stage) (von Auer, 1998)

4 It must be proven that ( ), ( ),

l lb bf S z A q is strictly negative in order to show that the effect of a more “dangerous” competitor

increases the risk of gathering resources.

6

With the given illustration, we can formally define the set of strategies available for both players

as such with their respective payoffs

For a two stage game, we have

[ , ] , [ , ]

[ , ]

[ , ] '

a r

b

a

For a three stage game, we have

[ , , ] , [ , , ]

[ , , ] , [ , , ]

[ , , ] , [ , , ]

[ , , ] '

[ , , ] '

a r

b r

c r

a

b

Where = {U [η, η] } = 0 when the player chooses a hide-hide strategy at any stage of the game.

In the case of a z-stage game, there are ( )zf variations of choices from the initial choice

made at the first stage, where z denotes the number of time periods and f is some payoff. Note

that for an agent l who decided a [ , ] strategy, immediately that agent perishes on or before the

end z stage5.

Defining the Game Environment

A simple derivation may be employed in order to define the characteristics of the players in

the game. With simple calculus and algebraic techniques, we attempt to show that the change in

the player’s chance of survival is basically the inverse of all other players’ adaptation function.

Consider Player A as our representative agent which experiences over z stages over time. Also

consider k natural environments that we denote by ke , where ( 1,..., )k K and lb players

( 1,..., )l L wherein ,K L . For each k

( )kk e ke P d e

where ( )kd e denotes the disutility of Player A from the difficulty on coping with the harsh natural

environment caused by the dystopia (e.g., a storm), which is the negative of the Epstein Zin utility

function holding optimal survival z as well as z’ constant such that

5 See Appendix A for Payoff Matrix

7

( ) , ', ,kd e U z z

and for each l in the economy

( ), ( ),l l ll b b bb P f S z A q

which is the functional form for the representative competitor of Player A we call Player B. We can

then construct the environment profile Ez,t as

, { , }z t k lE e b

Assuming a two player and two nature conditions6 (favorable and unfavorable) game, without loss

of generality we construct the environment profile for the players as

2 1 2 2 22 1( ) 1 ( ) ( ), ( ),

tz e e e b bE P d e P d e P f S z A q (10)

We let be constant for now7.

Decision of players with fixed probabilities

As one can note, we can assume that the player does not create harm to itself (hence it is

not part of its own environment equation), thus we can generally say that our player just reacts to

its state-of-the-world. Taking its derivative with respect to time while holding randomness of

probabilities constant (for now)

2 1 1( ) ( ) ( ) ( ) ( )tzdE d d d dS dz d dA dq d

P d e d e d e P S z A qdt dt dt dt dz d dt dq d dt

(11)

We let 1'( ) 0d e and 2'( ) 0d e

( ) ( )tzdE dS dz d dA dq d

P S z A qdt dz d dt dq d dt

(12)

We then let z q , and given the condition that players are at the minimum level of consumption

for optimal survival

( ) ( )tzdE d dS dz dA dz

P S z A qdt dt dz d dz d

6 We do not include the learning curve Ω in this derivation, as it contains a special property wherein if Ω=0, the game turns into an infinitely

repeated game with a nested payoff matrix (See Appendix A). 7 We consider the death rate as constant for now, since we want to focus on the intertemporal change in the Environment on the survival and

adaptation of all agents (holding the natural environment as a given)

8

At the condition where all players are stable, we set the derivative to zero

( ) ( ) 0dS dA dz d

S z A q Pdz dz d dt

Simplifying

( ) ( ) 0dS dA dz

S z A q Pdz dz dt

Which simplifies to

( ) ( )dS dA

S z A zdt dt

Assume a discrete change in survival '( )dS

S zdt

over time

( )dA

S A zdt

(13)

The result derived in Equation (13) has some interesting properties that can be exploited.

An inverse relationship between survival and adaptation denotes that X will decrease as Y becomes

better and better at gathering resources and vice versa. Hence we can see that in this model both

players have a competitive relationship that exists as long as they are living in the same

environment. Since the learning curve merely makes adaptation easier for both players, we can set

the learning curve Ω merely as some multiplier that augments the abilities of players to gather

resources.8

Obtaining the second total differential of the environment profile with respect to time, we

obtain the following result

2 2 2 2( )

2 1 2 1 12 2 2 2

2 2

2 2

( ) ( ) ( ) ( ) ln ( '( )) ( )

( ) ( ) ( ) ( ) l

tz td E d d d d d

P d e d e d e d e P t d edt dt dt dt dt dt

d S dz d d A dq d dS dz d dA dq dP S z A q S z A q

dz d dt dq d dt dz d dt dq d dt

n ( '( ))P t

(14)

We can now have a graphic representation of the environment profile which holds the learning

curve and all risk probabilities as constant, which may be positive or negative. This implies that

the environment profile can either evolve or devolve over time, depending on the characteristic of

the relationship with the two different outcomes (if the difference between the two types of

8 This result is also implicitly shown in our payoff matrix (see Appendix A)

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environment outcomes are positive or negative, and if this difference would be significantly larger

than 2 2 2

1 2 2 2( ) ( ) ( ) ( ) ( ) ln ( '( ))

d d S dz d d A dq d dS dz d dA dq dd e P S z A q S z A q P t

dt dz d dt dq d dt dz d dt dq d dt

.

Figure 4. Environment profile with respect to time (holding the learning curve and changes in risk probabilities constant).

Given that players do not have a learning curve and a constant probability of risk, it can be

observe that the environment profile becomes exceedingly more difficult as time passes. This

result is intuitive as one can observe that the governing force in setting difficulty for the game

revolves around the fact that risk is present and as time passes, the environment profile becomes

riskier. This means that the equilibrium point may be unstable.

Decision of players with dynamic and stochastic probabilities

We return to the original environment profile function and zoom into its characteristics,

only this time we now treat λ as a function of t since it is a Lyapunov9 exponent, which captures

the randomness of occurrence of the probability P.

2 1 2 2 2

( ) ( ) ( )2 1( ) 1 ( ) ( ), ( ),

t

t t tz e e e b bE P d e P d e P f S z A q (15)

It is also important to note the comparative dynamics of the environment profile with respect to the

probability of random occurrence of the events λ

2 2 2 1 1 1 2 2

( ) ( ) ( ) ( ) ( )2 2 1 1( ) ( ) ln ( ) ( ) ln ( ) ( ) lntz t t t t t

e e e e e e e e

dEP d e d e P P P d e d e P P f S z A q P P

d

(16)

From Equation (16) we can observe that the derivative of randomness of occurrence of the

event denoted by λ derives ln P which means that P cannot be valued at zero (but can

9 The Lyapunov Exponent is defined as the measure of chaos in a random system, such that

1

1lim ln ( ( ))

T

Tt

g x tT

,

( ( )) ( 1)g x t x t characterized in Appendix B (Cencini, et al., 2010).

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asymptotically reach zero at the limit). This ensures that for all ke there must be a positive P. This

means that all events must have a nonnegative, non-zero probability of occurrence, no matter how

small, and that the sum of all events must equate to one. For simplicity, we assume that all events

have a random probability of occurring, however they all have the same rate, hence2 1e eP P . This

goes without saying that e1 and e2 are mutually exclusive events.

Returning to the comparative dynamics of the model, we look into the interaction of agents

and time by taking the total differential of the environment profile with respect to time in this

manner

( ) ( ) ( ) ( )2 2 2 1 1

( ) ( )

( ) ( ) ln ( ) ( ) ( ) ( ) ln ( )

( ) ( ) ( ) ( ) ln ( )

tz t t t t

t t

dE d d dP d e d e P P t d e P d e d e P P t

dt dt dt dt

dS dz d dA dq d dP S z A q f S z A q P P t

dz d dt dq d dt dt

(17)

Interestingly, ln '( )P t can be interpreted as the growth rate of λ as , which means that for all

simultaneously occurring events at the initial stage, we merely factor in the growth (or intensity)

of chaos as well as the derivative of the functional via chain rule.

Simplifying we get

( ) ( ) ( ) ( )2 2 2 1 1

( ) ( )

( ) ( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

tz t t t t

t t

dE d d dP d e d e P d e P d e d e P

dt dt dt dt

dS dz d dA dq d dP S z A q f S z A q P

dz d dt dq d dt dt

(18)

Factoring out the derivative of σ with respect to time and ( )tP

( )2 1

( )2 1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

tz t

t

dE dS dz dA dq dP d e d e S z A q

dt dz d dq d dt

d dd e P d e d e f S z A q

dt dt

At the condition z = q

( )2 1

( )2 1 2

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

tz t

t

dE dS dz dA dz dP d e d e S z A q

dt dz d dz d dt

d dd e P d e d e f S z A q

dt dt

Since, by property of independence of random events (for the environment), we let '( ) 0kd e ,

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( )

( )1 2

( ) ( )

( ) ( ) ( ) ( )

tz t

t

dE dS dA dz dP S z A q

dt dz dz d dt

dP d e d e f S z A q

dt

( )

( )1 2

'( ) '( )

( ) ( ) ' ( ) ( )

t

t

dS dA dzP S z A q

dz dz dt

dP d e d e f S z A q

dt

At the stability condition

( )

( )1 2

: ( ) ( )

( ) ( ) ( ) ( ) 0

tz t

t

dE dS dA dzP S z A q

dt dz dz dt

dP d e d e f S z A q

dt

2 21 2( ) ( ) ( ) ( ) ( ) ( )b b

dS d dAS z d e d e f S z A q A q

dt dt dt

Holding the change in survival over time '( )dS

S zdt

at some discrete S changing over time

2 2 21 2( ) ( ) ( ) ( ) ( )b b b

dA dS d e d e f S z A q A q

dt dt

(19)

Let 21 2( ) ( ) ( ) ( )b bd e d e f S z A q

where reflects a linear sum all disutility factors for

Player A we get

( )dA d

S A qdt dt

(20)

Similar to equation (13), equation (20) has some interesting properties that can be observed, one

of which is the fact that the change in the chance of survival is again inversely related to all other

player’s change in adaptation, as well as the risk of independent death due to the choice to seek

resources, denoted by Λ. Moreover it could be proven that the second order derivative is positive,

as with equation (16)10

.

10

Similar to the fixed risks, the second derivative is also positive. Intuitively, this is due to the implicit risk from .

12

In the same manner as derived in equation (20), we can derive a complete characterization

of the survival function which now includes the player’s learning curve. For now we let the

learning curve be factored in the environment profile as the characteristic of Player B (which goes

without saying that if we allow for the learning curve to be factored in, everyone in the game also

has a distinct learning curve, including Player A)

( )dA d d

S A qdt dt dt

(21)

For both equations (20) and (21) however, the interesting characteristic these models is

how is also inversely related to survival. This means that Player A merely treats the

randomness of risk as a growth factor to the linear sum of its functional disutility derived from all

other factors. Simply speaking, Player A merely factors in the growth probability to the sum of all

his or her fears in surviving the dystopic economy. Hence Player A is able to simplify his or her

decision of a complex, stochastic environment into relatively more linear function that is simpler

to understand.

Fundamental behavior of players in the dystopic environment

Given the setup from our definition of the game environment, we can find interesting

properties which we can utilize in order to characterize the decision making of players in our

dystopic economy. We begin by proposing a condition which will (for sure) let our player’s optimal

strategy be to seek resources.

From the condition derived in Equation (34), we have characterized the relationship

between the survival function and the adaptation function. For the sake of simplicity of notation,

we denote Player A as X and Player B as Y. We can then verify that there exists a specific survival

function 1( )S z that intersects ( )XA z q as well as a specific survival function 2( )S z at ( )Yl

A z q

which we can logically imply that 1 2( ) ( )S z S z for all ( ) ( ) 0X Yl

A z q A z q . Now we can

graphically represent these functions accordingly

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Figure 5. First case; when ( ) ( ) 0X lY

A z q A z q and obtain their respective maximum at the same time t*. Note that

if the player chooses not to seek resources, Z=0.

Figure 6. Second case; when AX and AY do not obtain their respective maximum at the same time t*. Note that if the player

chooses not to seek resources, Z=0.

Proposition 1: (Strong axiom of revealed preference to seek)

Consider the cases where ( ) ( ) 0X lY

A z q A z q and obtain their respective maximum at the same

time t* and when AX and AY do not obtain their respective maximum at the same time t*. ( )XA z q

is a strictly concave production function with an existing maximum at q* for all q*>qb. A player will

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choose to seek for any given period t if and only if ( ) ( )X t Y tA q A q for all 0 given that the

Epstein-Zin survival function at a static point is Leonteif11

Proof:

Since the Leonteif function represents the players’ survival function, Player X will have a higher

payoff if he or she chooses to seek resources, due to the simple fact that his or her adaptability

function yields a higher payoff relative to Player B at t* since 1 1

Y Xt tz z and

2 2

X Yt tz z .

Taking advantage of the dynamic stochastic environment profile

A given decision based on a per-period advantage is at most trivial to characterize, which is

no different from what has been provided by the literature insofar as competitive games are

concerned (von Neumann & Morgenstern [1944], Maschler, M., Solan, E., & Zamir, S. [2013]).

However in a scenario wherein the respective probabilities of events are reasonably random,

(Shapley, 1953) decisions factor in these odds in order to maximize the payoff obtained. Hence, we

can utilize this randomness to our advantage to obtain a new theorem which generalizes the

decision of players in this specific kill-or-be-killed game taking place.

Figure 7. Set diagram for Player A (X) and Player B (Y), wherein the areas represent the “presence” of each player in the

economy, denoted by each of their respective V and V in neighborhood S

We describe the open set X as the safe zone in which player X uses in order to hide,

consequently reducing the risk of death from the outside world. At the boundary point of set X

supX we are able to seek resources vital for the survival of player X (since it is the least upper

11

We set the survival as some Leonteif since a static Epstein-Zin with already optimal consumption will yield a choice that

simply allocates all extra resources as consumption for the next period.

15

bound in which player X is at). We assume that in order to obtain max ( )Xq

A q we must be at least at

the boundary point, since all elements of the open set X are consumed by our player X (think of the

open set as the safehouse of X; due to the nature of our game we consume all minimum resources

in order to survive per period and stockpile for at least one period at a time).

At the boundary point, say A*, we are now exposed to risk imposed by our environment

profile. With that, we can now denote neighborhood S as the area inside the open ball with

radius r such that ( *, ) \ ( )B A r S X S which shares a risk probability from both players X and Y

such that for all ( )XA z q corresponds to some , denoting the elasticity to risk of X; for

1

( )Player Bl

l

A z q

corresponds to which denotes the elasticity to risk of Y (see figure 7). This will

also imply that there are different types of players with different levels of skills. Now, given that X

will encounter some specific individual at some time t, and if we do not assume that all other

factors remain constant, the probability that X will win or defeat the opponent at a specific

encounter is not necessarily the case, since we also have to consider the reaction function of X

towards the behavior (rather, the rationalization of the behavior) of the risk probability function

. Now, we can consider the second proposition.

Proposition 2: (Weak axiom of revealed preference to seek)

Suppose ( )XA z q is a strictly concave production function with an existing maximum at q* and

there exists a that maps the elasticity of Player A at V and an which maps the elasticity of

Player B at V in Neighborhood S. As long as X Y

while is positive, then Player A will

choose to seek for the period z for all 0 if and only if each ( )lY

A z q is distinct.

Proof:

As Yl approaches infinity with different characteristics, then if all of these Y’s will simply eliminate

each other except for the strongest player Y, but the probability of encountering the strongest

player Y inside the open ball is increasingly small as the number of opponents approach an

infinite number [see lemma 1], hence 1

lim 0L

ll

l

(which is similar to letting the radius of the

open ball expand infinitely). This means all players in the game will consider each other as a

threat, but consider as well that it changes as other players expand their r. Hence if we suggest

that r expand that it would imply that the environment profile

2 1 2 2 2

( ) ( ) ( )2 1( ) 1 ( ) ( ), ( ),

t

t t tz e e e b bE P d e P d e P f S z A q

will also expand the number of e1 will

also expand, which also implies that as the r increases, then even if you are the strongest, you, who

may have some form of weakness based on the environment profile: and conversely if you are

intrinsically weak, that would not guarantee that you will win in every encounter. This is based on

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the fact that the probability of survival given a specific skill set increases is not necessarily true,

since there is a random probability that a specific characteristic that X has is exactly what Y finds

to be part of his or her , which has a negative relationship to the environment. This also

affects the relationship between the probability that one will survive and win an encounter for the

next period given this intrinsic skill set that one has.12

Lemma 1

As Yl approaches infinity with different characteristics, then if all of these Y’s will simply eliminate

each other except for the strongest player Y, but the probability of encountering the strongest

player Y inside the open ball is increasingly small as the number of opponents approach an

infinite number.

Proof:

Let the set Y, 1 2 3

{ ( ), ( ), ( ),..., ( )}lY Y Y YY A q A q A q A q , l approaches , and ( ) ( )

i jY YA q A q ,

, 1,2,...,i j t . From the environment profile constructed (15) we can see that there exists a

corresponding ( ) ( ), ( ),

l l l

tl b b bb P f S z A q with a given

( )

l

tbP

which has the range (0,1). We set

2

( )1( )t

eP Y as the probability of player X to encounter the strongest player Y. By Bayes’ Theorem,

2 2 2

2

2 2 2

( ) ( ) ( )1 1 1( )

1( ) ( ) ( )

1 1

( ) ( ) ( | )( | )

( ) ( ) ( | )

t t te e et

e L Lt t t

e l e l e ll l

P Y X P Y P X YP Y X

P Y X P Y P X Y

Interestingly, as 2 2

2 2

( ) ( )1 1

( ) ( )

1

( ) ( | )lim 0

( ) ( | )

t te e

LL t t

e l e ll

P Y P X Y

P Y P X Y

since the denominator will approach infinity.

3. Conclusion and Discussion

From the previous construction, characterization and proposed strategies extracted from

the theoretical discussion, we have obtained the necessary and sufficient conditions for an

individual player who exists in an apocalyptic dystopia. With the corresponding relationship of

survival and adaptation, we have constructed a fundamentally understandable and intuitive model

that determines the effects of specific disutility functions, the adaptation level of competing

players as well as independent death rate to the chances that our representative player, Player A

(we denote as X), will face in order to survive the unforgiving landscape shaped by some

catastrophic event which renders conventional markets to freeze.

12

One can interpret this as the preparedness of an individual to the random probability of encountering an environment

with a heterogeneous set of natural hazards as well as enemy players.

17

First and foremost we illustrated a game tree that shows the decision path that Player A

takes as he or she traverses the dystopic economy. In this game tree, we have obtained a pattern

which can be characterized as an infinite repeating game as long as our Player A survives and is

given both options available; else if he or she chooses to hide the only logical action that Player A

can do for the next period is to seek resources.

Expanding the interactions to a two-player, two-environment game, we observe an

interesting phenomenon in which as long as Player A chooses a completely opposite strategy to the

opponent Player B, the two players will be able to maximize his or her chance to survive. This may

seem to be naively intuitive, however it is interesting to note that this is caused by having a less

complicated environment profile to face, since the environment profile consists of both the natural

hazard ke as well as the competing players in the game lb . This conclusion can be drawn from the

payoff matrix (see Appendix A).

We then zoomed into the more interesting phenomenon of the dystopic economy in which

we characterized the conditions of changes in the rate of survival when Player A and Player B do

meet. This will result to a clash and eventually a point in time wherein both players will fight over

the resource available in some time t. We derived these conditions for both the fixed risk case as

well as the dynamic stochastic case in order to maximize the time in which Player A survives the

apocalyptic event. In a nutshell, we have obtained that the relationship for both the fixed and time

varying, random probabilities are similar, except for the fact that for the dynamic and stochastic

case, we can note that the individual includes the growth rate of the probability that choice would

be risky or not as a weight in a weighted sum of all of his or her factors of disutility.

Given these characteristics of our players in the game, we have given two conditions

wherein the optimal strategy of X is to seek resources; hence we introduced the Strong and Weak

axioms of Revealed Preferences to Seek resources. These propositions have been given simple

proofs, verifying that when the model constructed is true, we are able to create a clearer picture on

what would be the optimal strategy of the representative Player A would choose given the option of

either to hide or to seek resources. These propositions are vital to policy makers, as these

conditions explain the behavior of the players in a dystopic economy.

One of the most important elements of the study is the realize this that whether or not you

are the strongest in a specific area, it is possible for you to lose, since by Proposition 2, even when

in terms of skills, the odds are stacked against you, the reaction function of Player A to the

environment must be taken into consideration, which may make or break the odds that Player A

will be able to live another day. With this we see that preparedness can be very crucial to

increasing the chance of an individual to survive and should play a significant role in

understanding the role of strategic planning and choice optimization. Given this fact, preparedness

for random events would be an essential element in future studies to be conducted to explore this

portion of the analysis. With this we can identify three key elements of survival in such an

economy; the environment profile, the level of skills that one must have, and the level of

preparedness to face the uncertainties of this kind of life.

The application of this theoretical game varies, when given a similar environment profile

as we have presented; from an actual survival strategy at some hypothetical fallout, or with

18

specific examples of chaotic and hazardous environments such as but not limited to aftermaths of

civil war, biological epidemics, massive prolonged technological blackouts, or similar disastrous

phenomenon. For future studies, it would be beneficial to measure the predictability of the model

constructed in this paper via simulations. This will aid policy makers in understanding the

behavior of agents and their respective reactions to the environment profile that the economy

would face given such unfortunate but reasonably possible circumstances.

19

4. REFERENCES

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Schuster, Inc.

Epstein, L., & Zin, S. (1990). First-order risk aversion and the equity premium puzzle. Journal of Monetary

Economics, 387-407.

Epstein, R. (2009). The Theory of Gambling and Statistical Logic. London: Elsevier, Inc.

Gali, J. (2008). Monetary Policy, Inflation and the Business Cycle. Princeton University Press.

Hughes, A., & Hughes, A. (Directors). (2010). Book of Eli [Motion Picture].

Konstantopoulos, T. (2006). Notes on Survival Models. Edinburgh: Heriot - Watt University.

Krusell, P. (2014). Real Macroeconomic Theory.

Luckerson, V. (2014, December 2). 5 Very Smart People Who Think Artificial Intelligence Could Bring the

Apocalypse. Retrieved January 13, 2016, from Time Website: http://time.com/3614349/artificial-

intelligence-singularity-stephen-hawking-elon-musk/

Maschler, M., Solan, E., & Zamir, S. (2013). Game Theory. New York: Cambridge University Press.

Shapley, L. (1953). Stochastic Games. Mathematics: LS Shapley, 1095 - 1100.

Stewart, W. (2009). Probability, Markov Chains, Queues, and Simulation. New Jersey: Princeton University

Press.

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Retrieved January 13, 2016, from The UN Refugee Agency Website:

http://data.unhcr.org/syrianrefugees/syria.php

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http://www.un.org/en/sections/about-un/overview/index.html

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http://www.un.org/en/sections/about-un/overview/index.html

Varian, H. (1992). Microeconomic Analysis. New York, NY.: Norton & Company, Inc.

von Aurer, L. (1998). Dynamic Preferences, Choice Mechanics and Welfare. Berlin: Springer-Verlag.

von Neumann, J., & Morgenstern. (1944). Theory of Games and Economic Behavior. New Jersey: Princeton

University Press.

20

Walpole, R., et al. (2012). Probability and Statistics for Engineers and Scientists. Boston: Pearson Education,

Inc.

21

Appendix A. Payoff Matrix for dystopian Hide and Seek game

Computation for payoff

N = Endowment = 1 unit at Stage 1

C = consumption = 1 unit per period

I = "investment" at "Seek" = 2

"Hide" = 0

Stage z = 1

Stage z = 2

Player A\Player B Hide Seek

Hide

Seek

12 ,P E r

1, 2r P E

2 22 ,2P E P E

0,0

Player A\Player B Hide, Hide Hide, Seek Seek, Hide Seek, Seek

Hide, Hide

Hide, Seek

Seek, Hide

Seek, Seek

1 ,P E r

1,r P E

2 2,P E P E

,r r

1 ,P E r

1,r P E

1 1,P E P E

1 1,P E P E

2 2,P E P E

13 ,P E r 1 2 2(2 ),P E E P E

1, 3r P E

2 1 2, (2 )P E P E E

2 1 2(2 ),P E E P E

2 2 1, (2 )P E P E E

2 23 , 3P E P E

22

Stage z = 3

Results:

The functional which denotes the learning curve of individuals at the dystopic economy has the following

properties:

1. At the 1,2 1,20

lim ( ) 0P E f

asymptotically, we can say that for all z = 1,2,..., there is no

difference between a 2 stage, 3 stage up to n stage strategy

2. Player A’s strategy will equal to Player B’s strategy if and only if they choose the opposite action of

the other player. Hence their strategies do not really matter despite the intrinsic chaotic environment

endogenous to the model.

3. Game is Nested to z = 2

Player A\Player B Hide, Hide, Hide Hide, Hide, Seek (impossible strategy)Seek, Hide, Hide Hide, Seek, Hide Seek, Seek, Hide Hide, Seek, Seek Seek, Hide, Seek Seek, Seek, Seek

Hide, Hide, Hide

Hide, Hide, Seek (impossible strategy)

Seek, Hide, Hide

Hide, Seek, Hide

Seek, Seek, Hide

Hide, Seek, Seek

Seek, Hide, Seek

Seek, Seek, Seek

,r r ,r r ,r r

,r r ,r r ,r r

,r r ,r r ,r r

1 1(1 ),P E r 1 1(1 ),P E r 2 1

2 1

(1 ),

(1 )

P E

P E

1 1, (1 )r P E

1 1, (1 )r P E

1 1, (1 )r P E

1 13 (1 ),P E r

1 1(1 ),P E r

1 13 (1 ),P E r 2 1 12 (1 ) ,P E E r 2 1 1

2 1

(2 ) ,

(1 )

P E E

P E

1 13 (1 ),P E r

1 13 (1 ),P E r

2 1 12 (1 ) ,P E E r

2 1 1

2 1

(2 ) ,

(1 )

P E E

P E

2 1 1

2 1 1

3 ,

3

P E E

P E E

1 2(1 ),P E r 1 2(1 ),P E r 1 2(1 ),P E r 2 1 2

2 1

,

(1 )

P E E

P E

2 1 2

2 1 1

,

(2 )

P E E

P E E

2 2

2 2

(1 ),

(1 )

P E

P E

1 2, (1 )r P E

1 2, (1 )r P E

1 2, (1 )r P E

2 1

2 1 2

(1 ),P E

P E E

2 1 1

2 1 2

(2 ) ,P E E

P E E

1 2(1 ),P E r 1 2(1 ),P E r 2 2 1(2 ) ,P E E r 1 2

1 1

2 ,

(1 )

P E

P E

2 1 2

2 1 1

2 (1 ) ,

2 (1 )

P E E

P E E

1 2 2

1 2 2

(2 ),P E E

P E E

2 2 1

2 2 1

(2 ) ,

(2 )

P E E

P E E

1 2 2

1 2 2

,

(2 )

P E E

P E E

2 1 1

2 1 2

2 (1 ) ,

2 (1 )

P E E

P E E

1 1

1 2

(1 ),

2

P E

P E

1 2, (1 )r P E

1 2, (1 )r P E

1 2, (1 )r P E

1 2(3 ),P E r 1 2(3 ),P E r 1 2 2(2 ) ,P E E r

1 1 2

2 1 1

2 (1 ) ,

)

P E E

P E E

2 1 2

2 1 1

3 ) ,

3 )

P E E

P E E

1 2 2

2 2

2 (1 ) ,

(1 )

P E E

P E

2 2 1

2 2 1

(2 ) ,

(2 )

P E E

P E E

1 2

1 2

(3 ),

(3 )

P E

P E

2 2 1

2 2 1

(2 ) ,

(2 )

P E E

P E E

2 2

1 2 2

(1 ) ,

2 (1 )

P E

P E E

2 1 1

2 1 2

3 ) ,

3 )

P E E

P E E

2 1 1

1 1 2

) ,

2 (1 )

P E E

P E E

1 2 2, (2 )r P E E

1 2, (3 )r P E

1 2, (3 )r P E

23

Appendix B. Characterization of the Lyapunov Exponent

Note: x implies the characteristic of the random event (in our case, the uncertainty from the

dystopic environment).

From 1

1lim ln ( ( ))

T

Tt

g x tT

and defining ( ( )) ( 1)g x t x t we obtain

ln ( ( )) ln( ( 1)) ln( ( ))g x t x t x t

Taking the absolute value of ( ( ))g x t ;

ln ( ( )) ln ( ( 1)) ln ( ( ))g x t x t x t

Now, taking 1

ln ( ( ))T

t

g x t

1

ln ( ( )) ln (2) ln (1)

ln (4) ln (3) ... ln ( 1) ln ( )

T

t

g x t x x

x x x T x T

ln (1) ln ( 1)x x T

Now, taking 1

1lim ln ( ( ))

T

Tt

g x tT

1

1 1lim ln ( ( )) lim ln (1) ln ( 1)

T

T Tt

g x t x x TT T

ln (1) ln ( 1)

lim limT T

x x T

T T

ln ( 1)

limT

x T

T

24

Appendix C. Alternative Proof for Proposition 1:

Since ( )XA z q has an existing maximum, we can map that some area U defined as the s

et of all possible positions where all players can be at some random point in time and let there be a

point of ( *)A q which denotes the function of survival of X and ( )bA q be some point where the

nearest Y’s to X could be. We now let S be the neighborhood of X within an open ball with radius

r such that ( *, )B A r S with infinite points of ( )bA q ; which means that there are an infinite

number of '( )bA q which correspond to some V that denotes to the change in survival of all Y’s

and some V which denotes the change in survival of X. This can be interpreted as the elasticity of

each player’s Arrow Pratt to the state-of-the-world (second order partial derivative). With that,

( ) ( )X Yl

A z q A z q implies that ( ) 1 0v v from the fact that the Arrow Pratt is twice

differentiable (hence existence of the neighborhood), therefore Player A is elastic to Player B’s

Arrow Pratt relative risk aversion, implying that the danger of Player A imposed to the

environment is greater than that of Player B. Hence Player A will prefer to seek for resources than

to hide. This is also due to the fact that the Leonteif yields a fixed minimum point.