Tuesday, March 15, 2016

Two Recent Deaths

We recently saw the deaths of two men whose impact on 20th century thought has been profound but relatively unsung. Philosopher and mathematician Hilary Putnam and mathematician/economist Lloyd Shapely were men whose penetrating and patient analysis provided the solution for many deep problems.

Hilary Putnam

Hilary Putnam was a student of the logical positivist Hans Reichenbach and from him brought many of the subjectivist, syntactical tools beloved of the Vienna Circle. However, his great mentor was the American philosopher Willard Quine. Like Quine, Putnam sought to use "pragmatism" to escape the subjectivist straight-jacket of the logical positivists. Pragmatism, in Putnam's sense, means that we commit ourselves to believe in the objects we talk about. Even if there is, in some "strictly physical" sense, only atoms & void/quantum fields/strings, this is only a factoid about some wacky definition of existence, not a deep fact about the world. In fact, we act as if we believe in tables, chairs, brains, minds, money and perhaps even justice. To a pragmatist, this "acting as if" is more real belief than philosophical handwaving. Putnam's great contribution in this area was also getting out of Quine's behaviorist straight-jacket. We act as if minds exist, so we ought to believe they do. Putnam argued that the mind could be understood as a cluster of machines in various states with complicated functions, and therefore could happily be allowed into a sensible world-view without inviting in spooks.

Through his interest in pragmatism, Putnam became very interested in forms of philosophy beyond the small analytical circle he was raised in/helped raise. The logical positivists were after an austere world view, a small number of subjective percepts could be held to exist and everything else was metaphysical. The American Pragmatists provided Putnam a way into a richer world. Through enriched world view, Putnam hoped to communicate with philosophical traditions ignored by pure positivism, even including religious thought.

This richer world extended into his teaching. Putnam's relative friendlyness to so-called metaphysics certainly provided his student David Lewis with a friendly place to develop his formal approach to metaphysics. Putnam's other students include the great logician and logicist George Boolos, the potent structuralist Paul Benacerraf and the pugnacious Jerry Fodor.

Hilary Putnam believe that philosophy should not be a quiet academic exercise separate from the sciences and from real life, and as befitting a pragmatist, lived out that belief. He was active in politics, played a key role in proving Hilbert's 10th problem undecidable and co-developed the logic checking album that was used to design the very computer you are reading this on.

Lloyd Shapely

The brilliant mathematician Lloyd Shapely was one of the fathers of cooperative game theory. In a cooperative game, groups of players can work together or against one another. In the board game Key To The Kingdom, the goal is to get the key and go to a specific square. All the players can cooperate to harry the player with the key - but such cooperation only works until one of them successfully steals the key...

It doesn't take much thought to realize that many situations are equivalent to cooperative games. Rousseau believed all of society was based on cooperative games. He asked us to imagine that we were a pack of men, women and/or wolves out to hunt meat. We could all eat hares separately and sleep with stomachs half empty. Or we could all work together and take down a stag and eat like kings. Akela, the wise wolf, wants his pack to be healthy and strong. How can he analyze this problem?

What he needs is a function from coalitions onto reward. Let's say that there are four wolves: Mom, Dad, Son, Daughter. We'll call this the set of players \( P \). Then the possible coalitions are:

$$ \emptyset \textrm{ - nobody}\\
\{ Mom \}, \{ Dad \}, \{ Son \}, \{ Daughter \} \textrm{one member}\\
\{ Mom, Dad \}, \{ Mom, Son \}, \{ Mom, Daughter \}, \{ Dad, Son \}, \{ Dad, Daughter \}, \{ Son, Daughter \} \textrm{two members}\\
 \{ Dad, Daughter, Son \}, \{ Mom, Son, Daughter \}, \{ Mom, Dad, Daughter \}, \{ Mom, Dad, Son \} \textrm{three members}\\
\{ Mom, Dad, Son, Daughter \} \textrm{four members} $$

This set is traditionally called \( 2^P \), or the power set of \( P \). The amount of delicious venison we eat is a function \( v: 2^P \mapsto \mathbb{R} \). This is called the Shapley Value Function. Analysis of this function allows Akela to organize the pack by - for instance - offering extra meat to those who contribute more, etc. With his infinite wisdom, Akela always finds a stable distribution. Note that these equilibrium distributions can obviously be unequal, as discussed before. For instance, if Father Wolf is strong and Mother Wolf is stealthy, Akela may deem it unnecessary to allocate extra meat to Daughter and Son Wolf.

The Shapley Value Function is very general. On the Wikipedia page linked above, it is used to construct a crude model of a firm. It can also be used to model a general market. The function can even be extended to when the set of players is very large so that no individual player has an effect on the whole game. This is modeled by making the set of players continuous. Integration theory then gives the fascinating result that the cooperative game optima are exactly the market optima! But this isn't just another way of writing Adam Smith's invisible hand. Combining continuous workers with discrete employers can become a model of oligopoly in hiring and much more besides. Such models are used by, for instance, Krugman & Fujita to model the spatial development of economies.

The fact that venison/utility can be easily reallocated between the wolves makes this game one of "transferable utility". Shapely also did foundational work in cases where "utility" cannot be transferred. Shapely famously this particular great contribution to economics on an afternoon dare. His colleague David Gale asked him if there was a way of marrying a number of men and women given their preferences and that the marriages had to be "stable" - no (man,woman) pair would divorce to get one another. Gale expected that if the number of men & women was large, then there would be no solution. After an afternoon's thought, Shapely showed that there always were completely stable patterns and gave an efficient algorithm for finding them! (allowing gay marriage makes the algorithm slightly more complex, but the result is the same)

Though stated in a silly way, this problem is very deep. Your computer bandwidth right now is allocated using this algorithm. The Gale-Shapely Algorithm brought Shapely into the world of "Mechanism Design", the development of markets and other methods for distributing in a stable way. This research would eventually net Shapely a Nobel Memorial Prize in Economics!

Anybody's death is a loss to humanity, but a part of them can live in our minds and our hearts. I hope that these men will be among those who will live in your mind forever.

No comments:

Post a Comment