12.26.10

Welcome back! It's time for Part 2 of:

Satan, Discount Factors, and Mortality's Grim Trigger: The Garden of Eden Game

Here we will be exploring the next Adam and Eve model which I have called the Latter-day Saint model (or LDS). This game has several differences from the OS model that I think you'll find interesting. And so, without further ado, here is it:

Model 2: Adam and Eve’s Forbidden Assurance Game

Again, this game has two players, but no exogenous players. We consider the following game.

		ADAM
		Eat	Don’t Eat
EVE	Eat	Ω, Ω	Ωε, βε
	Don’t Eat	βε, Ωε	β, β

Where Ω is the payoff for making covenants with the Lord, β, as before, is the payoff for staying in the Garden, and ε is the penalty to each player for leaving the Garden alone (Ωε) or failing to coordinate (βε). This is an assurance game and is also infinitely repeated. This means there is a discount factor δ here as well which, again, measures Adam and Eve’s knowledge of mortality. The principle payoff assumption is that Ω > β > Ω/β(ε) in the n-th game. That is, the eventual benefits of mortality outweigh the benefits to staying in the Garden, and that both these are preferred over any action with a coordination error term ε. This game also has an iteration where Adam and Eve become indifferent between equilibria as follows:

Ωε + Ωδ/1-δ ≥ β + βδ/1-δ

Where they, due to their understanding of mortality, realize that the payoff of making covenants outweighs the payoff of staying in the Garden combined with the penalty of leaving. Notice that there is no exogenous actor here and that the mortal stage of the game can be (was built to be) superior to the Garden of Eden stage—I discuss these assumptions and characteristics below.

A man after His own heart: The Plan of Salvation

The most obvious difference in the LDS game is that the payoff for making covenants in the mortal stage (Ω) is strictly preferred to the payoff for staying in the Garden (β), even in pure strategies. This traces to the fact that Adam and Eve’s understanding of covenants and the purposes of the second stage actually caused them to eat the fruit and leave the Garden. This is due to God’s intricate and complete outlining of the Plan of Salvation before the Garden of Eden or anything else was created. God did not intend for the Garden to be a permanent state of being for Adam or Eve; He therefore made it possible for them to learn the importance of mortality and thereby make the choice to suffer a small penalty (leaving the Garden) in order to receive the eventual payoff of mortal purpose.

Man does not live by bread alone: Complete Agency

Secondly, Satan has little power in this game except (perhaps counter-intuitively) to aid in Adam and Eve’s learning principles for themselves. Satan’s temptation of Eve actually sparked her contemplation of the purpose of the Tree of Knowledge of Good and Evil and she began to realize, as mentioned, the value of mortality. This implies the doctrine that Adam and Eve were subject to opposition in the Garden, but that their agency remained perfectly balanced. It also implies that Satan was actually the least knowledgeable actor in the narrative since he did not understand God’s purposes for mortality.

Time and tide wait for no man: Celestial Purposes

This leads to the last characteristic which is that of a complete agency. This type of agency allows the human family to have a progressive purpose which is not inferior to the divine. That is, because we have greater agency, our eternal potential is boundless. This traces to the doctrine of becoming like our Heavenly Father through the learning of a perfect application of our agency. 

Consider the following table which summarizes the characteristics and doctrinal principles discussed above:

GAME ASSUMPTION

1. The payoff Ω strictly dominates the payoff β in pure strategies

2. The discount factor δ is only manipulable endogenously

3. The grim trigger strategy is strictly preferred (Ωε outweighs β eventually)

CHARACTERISTIC

1. The eventual benefit of the second stage outweighs the Garden payoff, causing movement to mortality

2. Satan has no exogenous power over decision making

3. No inherent agency constraints meaning a more complete divine value

DOCTRINE

1. Plan of Salvation before the creation

2. Opposition to facilitate learning, but no agency constraints

3. Man’s agency and Celestial purpose and potential

Which, again, is pretty cool stuff. Again, this may not completely reflect the LDS point of view, but it does add some insight into a vastly different way of viewing the Garden of Eden. Interesting to think that such (seemingly) small actions by our first parents could trace to such important doctrines; this I will discuss a bit more in my next post. Hoped you enjoyed this one; stay tuned for the final post in this three-part adventure!

Part 3: Game Assumptions and Moving Mountains

Have a wonderful Christmas weekend! :)

12.20.10

This is Part 1 of a three-part post which I will entitle:

Satan, Discount Factors, and Mortality's Grim Trigger: The Garden of Eden Game

As the title suggests, as I was considering this semester what to write about as my final paper for my (exceptional) religion class, I decided that, as the class was about the Pearl of Great Price, it might be fun to do something about the Garden of Eden. And in this vein I thought: wouldn't it be fun to explore a bit more of the differences in doctrinal teachings of the Garden of Eden? And what better way to do so than to model them using game theory! After all, game theory is all about understanding why we make decisions and so, thought I, perhaps some of the vast differences we witness today in religious environs can be traced back to opinions and convictions about why Adam and Eve made their decision.

So, as Part 1 I have included here my first model which I have titled the Original Sin or OS model with an explanation. In Part 2 I will present and explain my second model which is called the Latter-day Saint or LDS model. And in Part 3, I will bring both models together and show how their subtle assumptions actually highlight some of the deepest doctrinal differences between major religions. Pretty fun stuff...well, at least I'm having fun, right?

Model 1: The Original Sin Prisoner's Dilemma

This game has two players (for those of you who don't get out much) with Satan as an exogenous player as follows:

		ADAM
		Eat	Don’t Eat
EVE	Eat	Λ, Λ	Π, β
	Don’t Eat	β, Π	β, β

Where Π is the fixed payoff for eating the Forbidden Fruit, β is the fixed payoff for staying in the Garden of Eden, and Λ represents the payoff for death (for moving to the mortal stage). You may notice that this is a prisoner’s dilemma and therefore, since the game is (assumed to be) infinitely repeated, I include a discount factor δ which denotes Adam and Eve’s knowledge about mortality (Λ). The main assumption of this game is that Π > β > Λ; that is, getting a one-period payoff from eating the fruit is preferred to a one-period payoff from staying in the Garden, which is preferred to death, ceteris paribus. However, to make Adam or Eve indifferent, Satan (the exogenous player) is assumed to have influenced both players such that:

Π + Λδ/1-δ ≥ β + βδ/1-δ

Or that they eventually reached a level at which they were indifferent between staying in the Garden and eating the fruit. This is assuming a grim trigger strategy since players can never return to Garden of Eden once they have eaten the fruit (i.e. they can never again reach the (β, β) Nash Equilibrium). The implications of this game will be discussed subsequently.

Better the devil you know: Adam and Eve’s lack of knowledge

The most obvious characteristic of the first model is that the payoffs are only loosely based on the second stage. This implies that mortality (the second stage) was seen as a situation of lesser value by the players of the game and by the game’s Creator. The Forbidden Fruit only existed as a facilitator of agency and the consequence of eating it was given as “death”. I argue here, however, that according to the OS model, neither Adam nor Even were much compelled to obey by the death penalty (no pun intended). And this again points back to the devalued view of mortality—the Creator did not want Adam and Eve to descend to such a state. This traces to the doctrine that, as mentioned in (reviewed) literature, the Garden of Eden was represented as the pinnacle of man’s existence and a paradise from which man was never intended to deviate.

An idle brain is the devil’s playground: Adam and Eve’s manipulation

According to the above assumption, the logic of Adam and Eve’s mistake must be explained exogenously. As implied, this makes Satan one of the prime actors of the Garden of Eden narrative. In an environment where Adam and Eve had little knowledge that extended outside their immediate surroundings, Satan was able to manipulation their perspectives in such a way as to convince that death was not a salient issue and that the payoff for eating the fruit was something of greater value than simply staying in the Garden. This traces directly to the doctrine of original sin—that each of us is born with inherently sinful natures that allow for a continual Satanic influence to permeate within us.

Between the devil and the deep blue sea: Why mortality is divinely sub-par

This discussion implies another fascinating principle: that Adam and Eve and, by inheritance, the whole human family has less divine value due to the Fall. A main assumption of the game is the grim trigger strategy—that players cannot again attain the Garden of Eden stage once the choice to eat the fruit has been made. Though heaven is the reward for the righteous, original sin makes it so that we, in this life, actually have constraints on our agency. And, given this restriction, there is less of a purpose to life as the attainment of a divine state (heaven) will always be strictly dominated by the potential lost in the Garden of Eden. This implies a deeper principle which is that, eternally, players will always have a lesser value and a lesser potential than God—no being can again attain the purity and communion with the divine as was found in the Garden.

And so, to review:

GAME ASSUMPTION

1. There is a discount factor δ which measures understanding of the future

2. The discount factor δ can be manipulated exogenously

3. The game is infinitely repeated with mortality being a grim trigger strategy (β > Λ always)

CHARACTERISTIC

1. This denotes a vague, manipulable understanding of the consequence called “death”

2. Satan had considerable influence over Adam and Eve in terms of decision making

3. Due to the Fall and agency constraints in mortality, the human family has less divine value

DOCTRINE

1. Garden of Eden as paradise (maximum potential of human soul)

2. Original Sin and the sinful nature of mankind

3. Human potential to become beings eternally less than

Pretty cool, huh? I mean, I could be totally off, but I think it is at least interesting to see how perspectives of Adam and Eve's choices trace to such important and salient principles. In fact, as I will discuss in Part 3, these principles actually reference some of the most important questions of life. And so, even though major world religions may not believe these assumptions exactly, they do present us with an interesting view of how one might interpret the Garden of Eden story. Hope you enjoyed it.

And don't forget! Stay tuned for Part 2: The Forbidden Fruit Assurance Game :)

11.7.10

Sandwich.

11.2.10

Thought you guys might enjoy this :)

One (wannabe) economist’s approach to home teaching (hehe):

Home teaching: a conundrum familiar to all members of the LDS church and the cause of more than a few grey hairs to Bishops worldwide. And how to solve it? Perhaps game theory can provide some insights.

I have sat through a fair amount of priesthood lessons where beloved leaders have gone on for several minutes appealing to my conscience in an attempt to push me towards doing my home teaching. Sometimes it can even have the opposite effect. And so this Sunday as I was sitting in Elder’s Quorum it came as no surprise to once again hear the Bishop start talking about home teaching. However, this time, as my thoughts began to wander, I found myself thinking, “I wonder if guilt really is the most effective encouragement strategy.” And so, like an self-respecting nerd, I build the following model using game theory.

I figure that leaders use two main strategies: reporting (enforcing) and not asking (not enforcing) which I will denote as (R) and (D). Home teachers also have two strategies: teaching (T) and not teaching (N). The two-player game can be represented as follows:

		LEADERSHIP
		Reporting (R)	No reporting (D)
HOME TEACHERS	Teach (T)	a1, a2	b1, b2
	Don’t teach (N)	c1, c2	d1, d2

Now, given a rational choice assumption, I would expect the following relationships to hold:

a₁ > c₁

(teachers would rather have taught if asked to report)

d₁ ≥ b₁

(some teachers may be indifferent, but most would rather not do their home teaching if they don’t have to report)

b₂ > a₂

(leaders would rather not have to go through so much reporting if the teaching is getting done)

c₂ > d₂

(leaders prefer to be notified if teaching is not being done)

And so, assuming mixed strategies, let p = the probability of teaching and q = the probability of employing a reporting structure. So, solving for p:

a₂p + c₂ – c₂p = b₂p + d₂ – d₂p

p(a₂ – c₂ – b₂ + d₂) = d₂ – c₂

p = (d₂ – c₂)/(a₂ – c₂ – b₂ + d₂)

And for q:

a₁q + b₁ – b₁q = c₁q + d₁ – d₁q

q(a₁ – b₁ – c₁ + d₁) = d₁ – b₁

q = (d₁ – b₁)/(a₁ – b₁ – c₁ + d₁)

Shocking, I know. It would appear the appealing to guilt and other such emotions (increasing the costs of not teaching—i.e. decreasing c₁ and d₁) actually has no effect on the probability of teaching. In fact, it has some adverse effects according to this model:

1.      Since d₁ ≥ b₁, the teachers on the margin may actually get less of a payoff for teaching when no reporting is taking place.

2.      Since a –c₁ is in the denominator of q, using guilt tactics will actually decrease the probability of employing a reporting structure.

3.      Since neither c₁ nor d₁ factor into the probability of teaching (p), teaching in unaffected.

Now, you may be asking: then how do we increase the probability of teaching? And the answer is, not surprisingly, to increase the payoffs within reporting structures. That is, to give report-gatherers a higher payoff for collecting reports (i.e. increase a₂ and c₂), thereby increasing enforcement which will cause the probability of teaching to increase.

However, if leaders are keen on the increasing the costs of not teaching (decreasing c₁ and d₁), I would suggest public reporting. That is, asking members publicly if they have done their home teaching and, if not, if they have plans to do so. This would certainly decrease enforcement, but perhaps not by a significant amount since the costs there associated would be smaller. Another tactic with similar results might be to, every Sunday, announce the names of those that have not yet taught that month.

As far as increasing the payoffs associated with reporting, I would suggest playing the “reporting” strategy with probability 1 along with moment randomization. That is, announcing that report-gatherers will call once a week and then randomly selecting when they call. This would hopefully induce teachers to form strategy expectations while also washing out some avoidance strategies. Also, although a reward system for report-gatherers might not seem ethical, not all rewards need be material. Bishops might think of simply calling and thanking those report-gatherers who were faithful in calling every week. This would increase the benefits associated with reporting and would thereby increase enforcement—consequently improving the teaching dynamic of the ward.

Now, the current teaching dynamic of my ward, as reported on Sunday, is about 50%. What does this say about reporting structures? Although there are several combinations, I might suggest the following:

R if T	R if N	D if T	D if N
a2 = 3	c2 = 1	b2 = 4	d2 = 0

Substantively, this means that my ward gets a high payoff if teaching is done, but absolutely hates reporting when teaching has not been done—this implies high social sanctions (i.e. awkwardness) in regards to reporting or could imply that the costs of reporting are high in relation to the benefits of teaching. Either way, this (wannabe) economist would suggest increasing c₂ at all costs—report-gatherers must be at least indifferent in their report-gathering preferences (c₂ ≥ a₂) in order to maximize teaching.

Cheers.

:D