(this is a paper in process, with areas still needing expansion and further definition, but which is posted here in this early form for comment, correction, improvement).

The Mathematical Equations of Symbiotic Investment: Maximizing Total Wealth by Investing in Each Other

George Byron Koch


This paper describes an approach to game theory, particularly in relation to negotiations between opposing groups of human beings. It begins by establishing some foundational definitions and equations which will later be incorporated into the final equations, and are fundamental to understanding their results and implications.


Raw materials: any form of matter, whether rock, water, oxygen, made up of atoms of elements in various combinations. Most raw material comes from the earth.


Energy: in the normal sense as used in physics, regardless of form or source, therefore including sunlight, coal, oil, wood, wind. Most energy used on earth comes directly or indirectly from the sun.


Conversion: the application of energy to raw materials to change their form, whether as simple as reshaping clay or as complex as converting silicon into integrated circuits. It includes such examples as the growth of micro-organisms, plants (which convert raw materials into plant fiber), the making honey by bees or milk by cows, and the manufacture of stomach acid, antibodies, ceramics, cars, televisions, cities and space probes by human beings. It also includes the conversion of energy into higher orders of organization (encompassed by information theory), such as in human corporate behavior and management, scientific theories, mathematics, philosophy, legal and moral advancement (as in equal rights, protection, safety, welfare, etc).


Sustenance: the raw materials and energy used by a living organism sufficient to sustain life.


Wealth: as narrowly defined here, the conversion of raw materials into products which go beyond the requirements of mere sustenance, and usually persist independent of their creator(s). Typically this is a tangible good, such as a clay pot or a television, but also includes scientific theories, medical procedures, literature, art, strategies, processes and even organizational and moral advances. Because it persists independent of its creator, wealth can also accumulate across generations and even millennia.


Living organism: an organizing entity which uses conversion to sustains itself, and which replicates offspring like itself.[1]


Offspring: a living organism produced by some form of replication from a living organism. In many ways similar to wealth, but distinguished from it in several ways.


Entropy: a decrease in order in a system, with a consequent release of energy.


Enthalpy: an increase in order in a system, with a consequent expenditure of energy.


The utility product, P, for a living organism (as a simple sum of conversion products and expenditures and without moral overtones) is:


P = L(S+R) + WS(G+H+M+A) – C(An+Dn+Au+Du) – O


L(S+R) is the conversion by the entity to sustain life, including sustenance (S) and replication (R).


W is conversion which creates wealth.


S(M+H+G+A) is the expenditure of energy to sustain life. This includes both the conversion of food into bodily Maintenance, Healing and new Growth, and that necessary expenditure to Acquire food (itself a product of the conversion of raw materials plus energy, and more complex at higher orders of life).


C(An+Dn+Au+Du) is the cost of competition with other entities, whether as Aggressor (as a necessary expense or unnecessary, An or Au) or Defender (as a necessary expense or unnecessary, Dn or Du). Most living organisms expend energy in all four of these cost categories.


O is the cost of Other energy spent in the many activities which do not sustain life, compete, replicate life or create wealth. An example of this would be eating to excess, smoking, destroying property or innocent life, even spending time in ways which are neither productive nor rejuvenating. Some of these may be morally repugnant, others morally neutral. But by definition (see wealth) none of them are morally positive, nor do they produce useful products.


Thus, the utility product P - that value which persists independent of the living organism – is the sum of its offspring and its created wealth.


None of this is particularly revolutionary. It simply describes the use of raw materials and energy by living organisms to sustain life and to produce offspring and wealth. In the grand view of things, the universe is apparently primarily an entropic process with localized enthalpic processes. Notable among these localized enthalpic processes are living organisms, which create order by harnessing and expending energy to re-order the structure of raw materials, in order to sustain life and create offspring and wealth.


However, there are some significant differences among the various orders of living organisms, and these affect the relative magnitude of the terms that sum to the utility product, P.


The utility product Pg for a group of living organisms is the sum of the products of the individuals:


Pg = Pi





Pg = [L(S+R) + WS(G+H+M+A) – C(An+Dn+Au+Du) – O] i


where n is the total number of individual living organisms in the group.


Given this utility product equation, let’s consider its terms, and the implications for individual and group behavior, at various orders of living organisms as they face various environmental and competitive challenges.


For all life forms, the conversion into the life of the organism itself L(S) is offset by the cost of sustaining life and the cost of competition for raw materials, primarily food.


For microorganisms and plants, Pg = L(R). That is, the Product is conversion into offspring, with virtually all remaining conversion spent on sustaining life. Both wealth W and other O appear to be small or nonexistent.


For insects, O appears to be non-existent, but W is greater than zero in some insect groups. Ants, for instance, will build anthills, tunnel, and some will actually herd aphids and milk them. 


For animals, O begins to have significance (although we seem to observe it, we would struggle to define it. Cats play with mice before eating them. Orangutans tap their fingers and stare into space). W can also be greater than zero. Beavers and birds build homes that can be used by subsequent generations (though not necessarily with that conscious intent). Some birds use tools, and a least one group of crows seems to be able to make them.


Nevertheless, for most living organisms Pg = L(R). That is, they produce offspring as the sole product which persists outside of their own subsistence. Most of the rest of their lives are spent acquiring available food to sustain life (by gathering or hunting), and competing for available resources with other living organisms. This effort also defines their relationship with other living organisms: they eat them, or compete with them for available food, or are eaten by them. Organisms which they can overpower, and which are edible, they overpower and eat. Those which are of relatively equal or greater power, they compete with and fear, and they are often injured, killed or eaten by them.


This competition and consequent fear of “other” is very primitive, and is found nearly universally among living organisms. It is defined here to include any competition for available resources, but usually also includes an identification of that which is self and (usually) one’s family and tribe (broader aggregations), and that which is outside of self, family and tribe: that which is “other.”


[expand on this a bit]


When individuals or groups compete, they strategize in order to win, whether that is to consume the opponent, win a needed share of the limited available food supply, or simply escape being eaten. We are not privy to the thoughts of insects or animals as they strategize, but we can easily witness it in their competition. Strategy may even be present in plants and microorganisms; we just can’t prove it.[2]  But among all of these groups, there is vanishingly little of the creation of profound and enduring wealth, such as the development of agriculture, medicine or transportation, and so these individuals and groups must usually compete with the “others” to acquire from what is naturally available that which they need to survive and reproduce. Their strategy is based on the reality of limited available supply and virtually no accumulated wealth.


The development of game theory analyzed how humans strategize as they compete with each other, and how groups of humans strategize as they compete with other groups of humans. This is made more complex because they not only try to anticipate each others’ moves, but each thinks about what the other is thinking, and even that the other is thinking about what the other is thinking about their thinking, and on and on.


Negotiating with the opponent can produce a result at least as good as the result produced by not negotiating at all, primarily because any communication increases the information each opponent has about the other’s needs and thinking: therefore each opponent optimizes his strategy at least to the degree he could without communication, and often better.


However, this fundamental strategy assumes a limited supply of available resources (whether food or whatever), which the opponents will divide in some way. We strategize to maximize the portion of this limited available supply which we will win. It’s called “game theory” because it is the stuff of competitive games, like chess, football or war, in which there is a winner and a loser (in some games, like war, this is seldom absolute). In applying the equation,



Pg = [L(S+R) + WS(G+H+M+A) – C(An+Dn+Au+Du) – O] i



we can see its results in several different scenarios:


C is high when there is competition, and O is reduced. W may be greater than zero for either opponent if they create weapons or strategies to give advantage, but this limits W because of the cost of aggression and defense.


When there is little or no competition (if the group were in a lush paradise with an abundant supply of food and no competition for it), C would be zero and O or W could be high, perhaps both.


If a lot of effort were involved to gather and process the food, and there were no competition, S would be high, C would be zero, and O would be small. W might grow in order to create more effective means of gathering, processing or creating food.


Where there are scarce resources and significant competition for them, S is high, C is high, and O is small. W would grow with the creation of strategies and weapons for the competition and possibly to improve the conversion of the resources, but this is offset by the cost of an increased C. (Put each of these in plain English).


The above, with minor variations, are the scenarios typically assumed when discussing game theory and strategy.


But what if resources (raw materials and energy) are relatively abundant, or if strategy and processes are created to make them so? Effectively, if W can be made high, while minimizing O and especially C, there may even be a reduction in S.


Put in very simplistic terms, when nearby tribes stop warring with each other over the limited game and vegetables that grow naturally, and instead turn their energies to agriculture, they can produce food for both tribes that vastly exceeds the needs of both. This is indeed the history of the human race: our creative efforts in agriculture, food processing, transportation and storage have vanquished the limitations of the unassisted natural production of food by the earth. The United States alone produces enough food to feed the entire world. No one need starve. But our primitive competition with each other – acting as if we are still fighting, life and death, over a limited natural supply – has prevented that food from reaching hungry mouths.


We act as if the supply of food and resources is strictly limited and we must fight for our piece of it, but the truth is that we can and do create abundance: we make the supply itself larger by our creative efforts. Agriculture, processing, transportation and storage have vastly increased the available food produced on this planet. But we fight and negotiate still as if it is the old limited natural supply. But our competition and fear of “other” seem to have blinded us and made us stupid.


This realization (that human efforts can increase the supply of what is to be divided) should radically change everything, including the very goal of our strategy. Instead of focusing on L, S, C and O, (that is, believing that our success is by necessity at the expense of the other, and that we fight to get as much as we can of a limited supply), we should now focus on W, the creation of total wealth. This not only creates more medicines, buildings, clothing, food and all the other products of human invention and hard work, it also actually increases the abundance of usable natural resources for conversion into wealth (as in making arid deserts bloom).


Therefore, instead of competing with the “other,” we should strategize to increase their production of W, and they likewise should work to increase our production of W. This maximizes W and minimizes S and C for all of us. It is therefore in our best interest.


Again, put simply, if the energy and resources now wasted in war and defense, in suppression and resistance, were turned instead to the creation of wealth, both sides would grow in wealth and prosperity. Instead of fighting over what we erroneously believed was a limited supply, we now cooperate to create a vast supply for both of us: just as did the primitive tribes that turned from fighting to agriculture.


That is,


Pt = Pg1 + Pg2


and therefore, if the strategy of Group 1 is to increase the productivity of Group 2, and the strategy of Group 2 is to increase the productivity of Group 1, the total wealth of both will increase.


Putting this in very modern and practical terms, what if the leaders of Israel announced tomorrow that their strategy with the Palestinians had changed, and that now instead of focusing the vast abilities and energies of Israelis on controlling the Palestinians and protecting against attack, they said, “We now dedicate ourselves to the success, the education, the health and wealth of every Palestinian.”


And if the leaders of the Palestinian people said, “We now commit ourselves to the health, welfare, safety and success of the Israeli people.”


If both groups realized the wealth this would create, in medicine, transportation, food, clothing and every aspect of human relationship, they would choose this strategy quickly. The costs of aggression and defense for both sides would drop. Wealth would increase. That area of the world would quickly become the wealthiest, and the safest.

This does not become true simply because it is a happy idea or a "hope for the future," but rather because each side understands fundamentally that such a strategy is in its best interest. Instead of fearing the "other," the "opponent," and wasting the creative energy of both sides on aggression and defense, when each side invests in the success of the other, both sides become far wealthier than possible under any other circumstances.


[to be expanded]

orig. 9/7/02


[1] Issues of consciousness and what constitutes “life” in a philosophical sense are not addressed here, not because they are considered unmeaningful, but because they are unrelated to the propositions and results.


[2] At least, not yet. It may be that what we assume is Darwinian among bacterial responses to anitbiotics may be adaptive or strategic at a higher level of organization, or even at the individual level, and even these may all be false differentiations.