Offering of Indulgences: A Game Theory An Operations Research Approach.

Or: what happens when a math major ponders theology.

A few weeks ago, Fr. Robert gave a homily on the topic of indulgences: how to obtain them, the difference between partial and plenary, and offering them for the faithfully departed. And me being me, I began to wonder: what is the optimal way to offer indulgences to maximize grace through one's life?

First, let's define an indulgence. According to the Catechism of the Catholic Church, "an indulgence is a remission before God of the temporal punishment due to sins whose guilt has already been forgiven, which the faithful Christian who is duly disposed gains under certain prescribed conditions through the action of the Church which, as the minister of redemption, dispenses and applies with authority the treasury of the satisfactions of Christ and the saints."

It continues, distinguishing between partial and plenary, "an indulgence is partial or plenary according as it removes either part or all of the temporal punishment due to sin."

Lastly, the Catechism specifies, "the faithful can gain indulgences for themselves or apply them to the dead." (CCC 1471)

This last statement is what I would like to focus on, for the purpose of this discussion.

Why offer them for the dead?

The Catechism addresses the practice of offering an indulgence for the faithfully departed in paragraph 1479: "Since the faithful departed now being purified are also members of the same communion of saints, one way we can help them is to obtain indulgences for them, so that the temporal punishments due for their sins may be remitted."

And again, in 1032, "From the beginning the Church has honored the memory of the dead and offered prayers in suffrage for them, above all the Eucharistic sacrifice, so that, thus purified, they may attain the beatific vision of God. The Church also commends almsgiving, indulgences, and works of penance undertaken on behalf of the dead:

"Let us help and commemorate them. If Job's sons were purified by their father's sacrifice, why would we doubt that our offerings for the dead bring them some consolation? Let us not hesitate to help those who have died and to offer our prayers for them."

Lastly, in paragraph 1475, it is explained how the benefits of the holiness of one member of the church can benefit another: "In the communion of saints, 'a perennial link of charity exists between the faithful who have already reached their heavenly home, those who are expiating their sins in purgatory and those who are still pilgrims on earth. between them there is, too, an abundant exchange of all good things.' In this wonderful exchange, the holiness of one profits others, well beyond the harm that the sin of one could cause others. Thus recourse to the communion of saints lets the contrite sinner be more promptly and efficaciously purified of the punishments for sin."

So the teaching here is that by offering indulgences for the Church Suffering, we can help them to obtain their final glory, and there is the hope that, once they have achieved perfect unity with God, there holiness can then benefit us here on Earth (the reality, not the planet).

During the sermon, Fr. Robert mentioned that, since we cannot know who is in purgatory, he always offers the indulgences he earns for his closest relative that is in purgatory, and lets God sort out who that is. This brought to mind my method of offering indulgences: to offer them for the soul with the longest sentence. The idea here was that if I got a soul who would otherwise have a long temporal punishment out of purgatory, that soul would be more thankful, and thus be more likely to use his spiritual goods for my benefit here on Earth. But then I began to wonder. Is this really the best way to achieve optimal grace and spiritual benefit?

Assumptions (no, not August 15ths).

Let us began by making certain assumptions about indulgences and the Church Triumphant.

Firstly, let us assume that upon helping a soul obtain the beatific vision, that soul will be grateful and offer his share of the Church's Treasury (CCC 1476-1477) for our benefit. This will give us some spiritual benefit, B. Secondly, let us assume that B can vary depending on the length of time, T, of the soul's temporal punishment. That is, a soul who was to remain in purgatory for a very long time, who we help out, will be more grateful to us, and thus offer us a bigger portion of his spiritual goods than a soul who was not to serve for very long. If true, this would mean that B would be greater for helping a soul with a longer sentence. Now let us examine three possible cases for how B can vary between souls with different lengths of their temporal punishment.

1) Logarithmic or no change.

In this first case, a soul who has a long sentence would either not be any more grateful upon being helped out of purgatory by our good works, or his gratefulness would increase logarithmically in relation to his increased sentence. Since we are assuming a direct correlation between sentence length, gratefulness, and spiritual benefit, this would imply that B does not change, or increases logarithmically between souls with differing sentence lengths.

We could express B as:

B = c ln T, for the logarithmic case, or

B = c, for the constant case.

2) Linear.

In this second case, a soul's gratefulness for being helped from purgatory increases linearly with respect to the amount of time he was to spend in purgatory. That is, B increases linearly with respect to the length of time which we, through our indulgence-earning actions here on earth, remit from the soul's time in purgatory.

In this case, we would write

B = cT

3) Geometric or exponential.

In this last case, a soul's gratefulness, and thus our spiritual benefit, increases at a geometric rate or an exponential rate with respect to the amount of time by which we reduce the soul's temporal punishment.

In other words,

B = kT^c, i.e., B increases geometrically with respect to the temporal punishment of the soul, or

B = c^kT, or B increases exponentially with respect to time.

(For c, k positive.)

So the purpose of this exercise is to determine, for a given assumption about the relation between spiritual benefit and temporal punishment, what is the best strategy to maximize B with respect to indulgences earned during one's life? Essentially, we are balancing getting many less-grateful souls out of purgatory against getting fewer extremely-grateful souls.

Plenary vs. Partial.

A plenary indulgence, according to the Catechism, removes "all of the temporal punishment" due to sin. Basically, a plenary indulgence acts as a "get out of purgatory free" card, that will automatically attain the beatific vision for a given soul. In this context, it is clear that, for all but the most trivial case (that is, where there is no difference in B between souls with short or long sentences), it is most advantageous to offer indulgences to souls who have the longest temporal punishment. For this reason, we will instead focus our study on how best to offer partial indulgences.

Let us assume that a partial indulgence remits a constant, finite amount from a soul's temporal punishment, where the amount remitted is dependent upon the act that earned the indulgence. We will call the amount of remission R.

The Trivial (Selfish) Solution.

The most obvious solution to the indulgence-offering problem is to apply all indulgences for the remission of one's own temporal punishment, according to CCC 1471. However, while applying indulgences to oneself removes one's own current sentence in purgatory, it does nothing to remit the temporal punishment incurred by future sins. Obviously, this is a short term solution, where there is no long-term benefit to the indulgence. This strategy is only recommended if death is feared to be imminent.

We will assume that helping another soul attain heaven produces some long-term benefit. That is, the soul, now in the presence of God, will have an ongoing and lasting influence in our life. Essentially, we are assuming that B is not temporary. As such, offering indulgences for the Church Suffering will have a better long-term ROI, whereas applying them to ourselves is the best solution until we commit our next sin.

Let us further examine the long-term case.

B is Constant.

In this trivial case, the gratitude showed by a soul we have helped does not differ based on the length of the soul's temporal punishment. This may be the case if a soul does not know the period of its time in purgatory. For the trivial case, it is best to maximize the number of souls we have helped from purgatory, since the benefit we receive is the same for each one.

B Increases Logarithmically.

In the case that the spiritual benefit offered by the soul whose temporal punishment we have helped to absolve increases at a logarithmic rate with respect to the length of said punishment, it is clear that applying the indulgence to a soul with a longer sentence suffers from diminishing returns. That is, the benefit received from the soul does not increase at a comparable rate with the time. This is because,

x(c ln T) > c ln (xT), for x, c positive.

In other words, if I receive x partial indulgences, where each one will remit t amount of temporal punishment, I have the option of getting x souls with sentences of length t out of purgatory, or I can get one soul with sentence xt out of purgatory. Since the latter case suffers from diminishing returns, the first case provides the most benefit to me.

In general, B = ^x/_p c ln (^p/_x t),

where x is the number of indulgences obtained, p is the number of indulgences applied to each given soul, and t is the total amount of temporal punishment remitted by all received indulgences. A simple calculation will show that this is maximized at p = 1.

B Increases Linearly.

In the case that B increases in a linear relation to the amount of time, T, remitted by our indulgences, there is no difference between which souls we apply our indulgences to. This is because,

x (cT) = c (xT)

In other words, the benefit to us is the same, regardless to whether we help one soul, with temporal punishment xT or x souls, with temporal punishment T.

B Increases Geometrically or Exponentially.

In these last two cases, it is more beneficial to apply an indulgence to a soul with a long temporal punishment, as the spiritual benefit we receive will increase at a rate exceeding the temporal punishment we help to remit. This is because,

x (kT^c) < k(xT)^c

and,

x (c^kT) < c^kxT

For this strategy, it is best to offer partial indulgences for the soul with the longest time in purgatory, that we can remit during our lives.

Why's that?

It is important to note that even in the last case, it is not most beneficial to offer our indulgences for the soul with the longest temporal punishment. One must consider the possibility that the sum total of all partial indulgences we earn will not equal the sentence length of the soul we offer them for. In this case, as of the time of our death, we will not have elevated the soul to the status of Church Triumphant, and will not have begun to receive our spiritual benefit, B. This is why it is important to make sure we at least get one soul out of purgatory with our indulgences.

A Soul's Access to the Church's Treasury.

Another variable to take into account is the holiness of the souls we are helping. It is conceivable that a holier soul would have greater access to the spiritual goods of the Church's Treasury, and thus elevating a holier soul could have a greater benefit, even with less gratefulness on the part of the soul. In this case, even though soul A, whose temporal punishment, and therefore gratitude, is much greater than that of soul B, soul B may have much greater access to spiritual goods. 

In other words, we wish to maximize,

B' = B · H_s,

where B' is our adjusted spiritual benefit, B is the unadjusted benefit, derived from the gratefulness calculations above, and H_s is the holiness of the soul whose punishment we have remitted. In this case, reasonable assumptions must be made about the likely holiness of the souls.

This approach would seem to favor souls with shorter temporal punishments, as these souls are likely to be holier than those with longer temporal punishments.

NB: I could find no indication in the Catechism that a soul's access to spiritual treasures depends on that soul's holiness during his life. If this is the case, one can safely discount holiness as a factor in application of indulgences.

Spiritual Interest Rates.

One more thing which we ought to consider is the moral decay, or lack of spiritual advancement in our lives brought about by delayed spiritual benefits. Simply put, having more members of the Church Triumphant, even those with a small benefit, praying for us now may, in the long run, provide us with greater benefit than having one soul praying for us in fifty years. This can be thought of as "spiritual interest", and can be likened to banking interest rates. For instance, is it wiser to invest $2000 now, or $5,000 in ten years?

To take this into account, one can simply apply the compound interest formula to the spiritual benefit we've calculated previously. In other words, we want to maximize the total lifetime benefit, B_L, where,

B_L = H(1 + B/n)^nt,

where H is our initial holiness, B is the spiritual benefit that we expect to receive from our strategy, as calculated above, n is the number of souls we expect to elevate to Church Triumphant per year with our current strategy, and t is our lifetime.

Plenary vs. Partial, Part Deux.

One more thought on plenary vs. partial indulgences: in order to maximize spiritual benefit during your life, you should never offer a plenary indulgance for a soul for whom you have already offered a partial indulgence. In such a case, the partial indulgences you've already spent on the soul will in essence be wasted, since they did not serve to remit the soul's punishment any faster. Instead, you might offer a plenary indulgence for the soul for whom your worst enemy has spent the most partial indulgences on, to minimize the spiritual benefit gained by him.

Note that while I could not find any evidence in the Catechism to indicate so, common sense might indicate that such a strategy would cancel out any earned indulgences, as part of the requirement for earning a plenary indulgence is to be "free from any attachment to sin," and such spiteful application of an indulgence would indicate the opposite.

God Knows What's Best.

With all these variables, and special cases, and unknowns, it seems that it is very difficult to find the optimum indulgence-application strategy. However, God always knows what is best for us. It is for this reason that I recommend we fall back on God. When you offer an indulgence, you can simply make a prayer as such, "Father in Heaven; I offer up the graces I receive today to that poor soul in purgatory whose gratefulness will maximize the equation:

B_L = H(1 + B'/n)^nt,

where B_L is the total spiritual benefit achieved in my lifetime, H is my current holiness, n is the number of souls I will help out of purgatory in my lifetime, t is my lifetime, B' = B(T) · H_s is the adjusted holiness of the soul (where applicable), and B(T) is the gratefulness function for a soul, based on the time of his temporal punishment. Amen."

God will know what you mean.