Trembling Hands

At least since Selten (1975) game theorists have considered that given a series of decisions there is some small probability that the person making the decisions will make a mistake and do something irrational, even if she knows the right thing to do.  This is called the trembling hand approach: although a person rationally knows the right (rational) thing to do, sometimes her hand trembles and she chooses incorrectly.

Therefore, given a game defined by a finite set of iterated decisions and payoffs in which all the rational moves are known by both players (think Tic Tac Toe), there is a ‘perturbed’ game in which the rational choices are not made.  So consider playing a game of Tic Tac Toe:  Either player can always force a draw in Tic Tac Toe and hence prevent loss.  However, it is easy enough to make a mistake (through inattentiveness, eg) and allow your opponent to win.

Tic Tac Toe Game Tree

Tic Tac Toe Game Tree showing possible decisions for the first two moves

I believe this approach is a good start but does not go nearly far enough to incorporate probability into game theory.  The issue stems from the trembling hand approach assuming that irrational behavior occurs because of ‘some unspecified psychological mechanism.’  This is fine, but then every trembling hand probability, every chance of making an irrational decision, is defined as a separate, independent probability.  This means that making an irrational decision is based on chance, as if we roll a die every decision we make.

Perhaps some people have this problem, that they act irrationally at probabilistic rates, but this doesn’t seem either realistic, or fit with the idea that a psychological mechanism was at work.  If some psychological mechanism was at work, then we would expect

  1. The probabilities of making mistakes would not be independent of each other, since they have a common source.
  2. There would be a much higher chance of irrationality at times when the psychological issue manifests itself.

One example of what I have in mind is the effectiveness of gamesmanship in sport.  Gamesmanship is the art of getting into your opponents head and causing them to make mistakes.  Consider this description of “furbizia” in Italian soccer by Andrea Tallarita:

Perhaps nothing has been more influential in determining the popular perception of the Italian game than furbizia, the art of guile… The word ‘furbizia’ itself means guile, cunning or astuteness. It refers to a method which is often (and admittedly) rather sly, a not particularly by-the-book approach to the performative, tactical and psychological part of the game. Core to furbizia is that it is executed by means of stratagems which are available to all players on the pitch, not only to one team. What are these stratagems? Here are a few: tactical fouls, taking free kicks before the goalkeeper has finished positioning himself, time-wasting, physical or verbal provocation and all related psychological games, arguably even diving… Anyone can provoke an adversary, but it takes real guile (real furbizia) to find the weakest links in the other team’s psychology, then wear them out and bite them until something or someone gives in – all without ever breaking a single rule in the book of football. (via)

If we try to explain the an instance of someone making an irrational play in a game due to gamesmanship/furbizia according to the trembling hand model, we run into difficulty.  The decision tree according to the ‘trembling hand’ theory would have a series of decisions each with a low probability of making an irrational mistake:

.01 — .01 — .01 — .01 — .01

Hence it cannot explain why someone would crack later in the game as opposed to earlier, since all the probabilities are equal.  Nor can it explain why people make irrational decisions at higher rates when playing against a crafty opponent than they would make otherwise. Therefore the trembling hand model cannot explain the effectiveness of gamesmanship.

But the decision tree given linked, non-independent probabilities might have the chance of an irrational decision given by:

.01 — .05 — .1 — .17 — .25

This model has an increasing chance of irrational action.  As time progresses, it becomes increasingly likely that an irrational choice will occur due to the gamesmanship of the opponent.

I’ll refer to this model generally as induced irrationality.  Induced irrationality occurs when the chance of making a rational decision decreases due to some factor, or when the chances of making irrational decisions over time change in concert, or both.

Other phenomena follow this pattern.  Bullying comes to mind: it is similar to gamesmanship in its breaking or bending of ‘rules’ over time to get in someone’s head and thence get them to do things they would rather not do.  The bullied will act irrationally in the presence of the bully and potentially more so as the bullying continues, perhaps even leading to “snapping”— doing something seriously irrational.

Phobias are also similar: for whatever reason a person has a phobia, and given the presence of that object or situation, the otherwise rational person will make different decisions.

Moreover this may have something to do with the Gambler’s Fallacy:   By making a gambler associate a pattern to some random act, such as by showing the gambler all the recent values of a roulette wheel in order to convince the gambler to believe that the wheel likely will land on red (or losing a few bets to a shill in 3 card monte, or seeing a pattern in the stock market, etc.), the casino has planted a belief in the gambler.  Hence, as time goes on and red is not landed upon, the gambler increasingly thinks it is ever more likely that red will hit (even though it has the same low chance as it always did). Hence the gambler will likely bet more later — more irrationally —  as he expects red to be increasingly likely to hit.

Hence, though trembling hands may be a factor in irrational decision making, it does not seem like it is the only possibility or even the most significant in a number of interesting cases.


Selten, R. (1975). ‘Re-examination of the Perfectness Concept for Equilibrium Points in Extensive Games.’ International Journal of Game Theory, 4: 22–55.

My brother beat the Tic Tac Toe playing chicken when the Chinatown Fair Arcade (NYC) still operated.  I assume that there was a computer choosing the game moves and it happened to glitch when my brother was playing: though the machine claimed it won, if you looked at the Xs and Os, my brother had won.  We asked the manager for our promised bag of fortune cookies.  He said he didn’t actually have a bag since the chicken wasn’t ever supposed to lose.

Posted in economics, game theory. Tagged with , .

Holy crap, it’s The Matrix for chickens

No, really:

[Sent to me by my brother. Thanks bro!]

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The Rationality Cone

There are different sorts of constraints on thought.  We forget things, we fail to infer consequences of our beliefs and we have features of perception, like blind spots, that affect our understanding of our surroundings.  We also can be greatly affected by our emotions: when we are angry — when we see red — we are unable to see the anything but the things that are making us mad; when we are infatuated we are, conversely, unable to see anything wrong with our object of desire.

This account of emotional states — moods — is interesting because moods affect our overall reasoning ability.  Given one mood, we will be able to make certain inferences; given another, we will make different inferences.  Moreover, what seems to be a rational inference in one mood may be irrational in another mood.

At this point we have thought, which is comprised of our knowledge, perception and deductions we make, and mood, which modifies and constrains thought.  If we consider the situation over time, then at any point a person has a history of thoughts and moods, which has led up to the current state, and a potential future of moods and thoughts based on where that person is now.   Going from one mood to another, or one thought to another, can only be done within a limited range, i.e. no thoughts or moods can be completely detached from prior thoughts or moods.  This gives us a perspective on the relation between rationality, consciousness and thought:

Considering a person’s consciousness at some point, we can map what we consider rational and irrational based upon the potential mood and thought changes.  Any possible future belief (a combination of thought and mood) will be a combination of changes in prior moods and thoughts.  Beliefs that require too great a change in both thought or mood may be outside the realm of rationality for a person, while beliefs that require little effort will fall within the realm of rationality.  Hence, the rationality cone

Rationality Cone Diagram

The Future Rationality Cone illustrates how, given changes in thought or mood, a person’s beliefs can become different from their current beliefs.  The edge of the cone is the limit of what that person could possibly rationally believe:  anything outside the edge requires too great a jump in thought or mood from where they currently are.  Any point inside the cone represents a set of beliefs that the person could rationally have, given different circumstances.  The bottom half of diagram shows the past rational states that could have led up to the current state, as represented by the Past Rationality Cone.


If the above scheme is familiar, it is because it is modeled on the Light Cone from physics; the diagram is from the linked Wikipedia page.  I always found it fascinating that the light cone implies that there is part of the universe immediately surrounding each of us that we can never physically access.  Likewise, there are thoughts and moods that are just like our own that appear irrational to us—even if they are not—because they fall outside our capabilities.  Other people could, however, have these thoughts because their rationality cones are not exactly aligned with our own, or they started from another location, which enabled them to access that part of the mental universe.

Posted in metaphysics, mind, philosophy, physics.

Яandom Logic

If we try to represent tossing a coin or a die, or picking a card out of a deck at random, in logic, how should we do it?

Tossing a coin might look like:

Toss(coin) → (Heads or Tails)

Tossing a die might be:

Toss(die) → (1 or 2 or 3 or 4 or 5 or 6)

Picking a card:

Pick(52 card deck) → (1♣ or 2♣ or … or k♥)

This begs asking, do these statements make sense? For instance look what happens if we try to abstract:

∀x Toss(x)

such that ‘Toss’ represents a random selection of the given object.

But this is weird because Toss is a randomized function and x is not selected randomly in this formula. Perhaps if we added another variable, we could generate the right sort of function:

∀y ∃x Toss(yx)

Then x would be a function of y: we would select x with respect to y. The problem is still that a Toss involves randomness. So this setup is incorrect because treating x as a function of y is not randomized, because y is not random.

How can we represent randomness in logic?

As noted, functions alone will not work. Variables and interpreted objects cannot invoke randomness. Perhaps we can modify some part of our logic to accommodate randomness. The connectives for negation and conjunction haven’t anything to do with randomness either.

But, if we use the game theoretic interpretation of logic, then we can conceive of each quantifier as representing a player in a game. Players can be thought of as acting irrationally or randomly.

Therefore, let’s introduce a new quantifier: Я. Я is like the other quantifiers in that it instantiates a variable.

  1. Яx T(x)
  2. Tb

However, Я is out of our (or anyone’s) control. It does instantiate variables when it is it’s turn (just like other quantifiers) but it instantiates randomly. So we have three players, Abelard, Eloise and Random (or the Verifier, Falsifier and Randomizer).

But more is still needed. We need a random selection between specific options, be it between heads and tails, 1-6, cards, numbers, or anything else. One way of doing this would be to create a special domain just for the random choices. Я would only instantiate from this domain, and if there are multiple random selections, we will require multiple indexed domains.

Hence, given Di(Heads, Tails),
represents a coin flip since Я randomly instantiates out of the domain containing only Heads and Tails.

I prefer to use an artifact of Independence Friendly logic, the dependence indicator: a forward slash, /. The dependence indicator means that the quantifier only depends on those objects, variables, quantifiers or formulas specified. Hence

Яx/(Heads, Tails)

means that the variable x is randomly instantiated to Heads or Tails, since the only things that Яx is logically aware of are Heads and Tails. Therefore this too represents a coin flip, without having multiple domains.)

Now that we have an instantiation rule for Я we also need a negation rule for it. If some object is not selected at random, then it must have been individually selected. In this case the only other players that could have selected the object are ∀ and ∃. Hence the negation rule for Я is just like the negation rule for the other quantifiers: negating a quantifier means that a different player is responsible for instantiation of the variable. If neither player is responsible, it can be considered random: ¬Яx ↔ (∀x or ∃x). We can leave the basic negation rule for ∀ and ∃ the way it is.

Therefore, given the additions of the new quantifier and domain (or slash notation), we can represent randomness within logic.


See “Propositional Logics for Three” by Tulenheimo and Venema in Dialogues, Logics And Other Strange Things by Cedric Degremont (Editor) College Publications 2008, for a generalized framework for logics with 3 quantifiers. Since the above logic requires either indexed domains or dependence operators, Яandom Logic is a bit different, but it is a good discussion.

Posted in game theory, logic, science. Tagged with , , , , .

new york

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Book Review: The Genial Gene

The Genial Gene: Deconstructing Darwinian Selfishness by Joan Roughgarden

In The Genial Gene Joan Roughgarden seeks to replace the competitive understanding of evolution, known as sexual selection, with a cooperative one. The first sentence of her book reads, “This book is about whether selfishness and individuality, rather than kindness and cooperation, are basic to biological nature” (p. 1).

So what is the argument? Taking this first line, she wants to conclude something about basic biological nature. To do this, one can either define what basic biological nature is and then use that definition to derive conclusions, or else survey the natural world and find the best interpretation for whatever empirical results were found. She opts for the latter strategy.

To this end she first surveys and compiles examples of what people consider to be evidence for sexual selection and argues that this evidence has been misconstrued or simply does not support the theory of sexual selection. Then she offers a few logical arguments against sexual selection with the aim to highlight contradictions within the theory.

She then develops her alternative, called Social Selection. Social Selection is fundamentally based upon cooperation, not competition, and she proceeds to reinterpret the empirical research with respect to cooperation. Given the results of this reinterpretation, she concludes that the cooperative approach provides a more accurate picture of the empirical data than the competitive approach. Therefore social selection, not sexual selection, is fundamental to biological nature.

Can this argument be maintained?

Her argument fundamentally turns on the interpretation of the empirical research. (If her logical arguments were strong enough to undermine sexual selection on their own, she would have dedicated more space to them. At best, in my opinion, they could raise questions about sexual selection, but are not inherently damaging enough, even if they are accepted uncontested, to force a major revision to sexual selection.) She interprets the research in terms of cooperation and her opponents are those who interpret the research in terms of competition. Roughgarden claims her interpretation is the correct one.

Insofar as she is making an inference saying her interpretation is the best conclusion, her argument fails. She readily admits that the defenders of sexual selection are able to consistently create explanatory fixes for apparent contradictions in the empirical research. Since they are able to explain the data, the fact that she is unsatisfied by their explanations (and likes her own better) is no grounds for convincing her opponents to give up their explanations. After all, they have history and authority on their side. Her coming up with better numbers, that is, having formulas that (she says) more accurately represent the research, is not a sufficient reason for discarding a theory that has held up for many years, especially one that continues to be an area of active research. So, she has not successfully argued that social selection should replace sexual selection.

However, if we consider a more modest conclusion, then Roughgarden may be able to maintain part of her argument. She makes the point that the core Darwinian theory does not include sexual selection; it is a later contribution (ppg. 3-4). This suggest that there may be theoretical room for cooperation in addition to competition. But how much room?

Now the interpretive problem that she set up cuts the other way. Instead of her trying to convince us that her cooperative interpetation of the empirical research is the correct one, we ask the competitive interpretation why it is the best one. Empirical research alone cannot support one conclusion over another: the data must first be interpreted before a conclusion can be reached. As mentioned above, sexual selection has history and authority on its side, but age and endorsements are not arguments for being the sole fundamental methodology of biological nature. Without history and authority, sexual selection proponents only have their ability to explain bioogical research, which is no more than Roughgarden has. Therefore, advocates of sexual selection have no further theoretical resources to support their claim that sexual selection is the fundamental method working in evolution.

This means that Roughgarden does succeed in part. Based on the arguments she provides she is unable to maintain that kindness and cooperation underpin evolution, but she is able to cut sexual selection down to her size. She has shown that it is possible to reinterpret biological research in terms that do not rely upon competition and that sexual selection has no special theoretical privelege. Therefore sexual selection proponents can no longer claim to be fundamental biological reality: even though Roughgarden was unable to fell their theory, they won’t be able to down her either, and so she has established theoretical room for cooperation in Darwinian theory.

Posted in biology, evolution, measurement, science. Tagged with , , , .

Philosophy Carnival #2

Well, it’s the second philosophy carnival I’ve hosted.  So let’s get started.

Counterfactuals and time traveling cold-blooded murderers!  Why is it we always want to see what happens when we kill ourselves (or others) when time traveling?  Does time travel make one murderous?  Anyway, besides  the weird questions that occur to me, the discussion over at Kadri Vihvelin’s philosophy blog does try to tackle Counterfactuals, Indicatives and What Time Travelers Can’t Do.

If that isn’t your cup-o-tea, maybe you like smoking pipes.  But apparently not the pipe if you are from Utah.

And if you don’t like smoking pipes, nor counterfactual time travelers, then perhaps you like the movies.  Over at Pirates and Revolutionaries we have durationless movies that involve no time flow.    Lot’s of YouTube clips — which I did actually watch (most of them at least) and you should too. Because what good is anything without pictures or conversation?

Well, if you don’t like the movies, pipes or homicidal time travelers, try the news.  But not if you read the New York Times’ Stone, apparently, because over at The Consternation of Philosophy, Matt says they got the foundation of human rights wrong.  He writes that the reasons that are given not only do not show a foundation for secular human rights, they actually lead to dogmatism.

But maybe you like dogmatism.  I bet there are secretly lots of philosophers out there who, while publicly decrying dogmatism, are secretly delighted with their own.  I’m probably one of these people.  Then, perhaps we should be epistemological anarchists, as is suggested over at the Kindly Ones.  Paul writes on Feyerabend’s reductio directed at rationalist conceptions of scientific method, which concludes with: Anything Goes.  You should read this- it’s something I dogmatically recommend.

Fine, be that way.  Don’t do what I recommend.  Instead sit around and listen to the radio for all I care.  Actually, we’ve got some high quality internet radio going down at the Partially Examined Life: Pat Churchland on the Neurobiology of Morality (Plus Hume’s Ethics).  I surprised myself and listened to the whole thing.  Good talk.  Also, a book for sale.

If none of this armchair stuff has impressed you, I guess you might be one of those “go out and do stuff types.” [yeah right]  If that is the case, go check out some Experimental Philosophy.  Justin writes, ask not “what the history of philosophy can do for us, but rather what we can do for the history of philosophy.”  That’s the attitude!  (The post is actually on the place of x-phi in a historical and cultural setting.  Good stuff just the same.)

Or maybe you are just at The Ends of Thought, so frustrated with the difficulties that you’re left wondering where a lot of us went wrong (and I’m not saying who has gone wrong; you’ll have to read the post to see who Roman says did).

Lastly, zombies, because philosophers like zombies.

Posted in fun, philosophy.

Philosophy Carnival 8 August

I’m hosting the next philosophy carnival, on August 8th.  If  anyone is thinking about submitting to the next philosophy carnival, I have a preference for philosophy of science, though feel free to submit on any topic.  Also, I like to be entertained by my academic philosophy, so the more off the wall the better.

Posted in news, philosophy. Tagged with , .

The SEP on the Kindle

Reading on a computer screen is often not pleasant, especially when a lot of reading has to  be done. This is a general problem for philosophy since nearly everything is in PDF format and if you don’t want to print out a tree’s worth of paper you are stuck.

I got a Kindle.  Kindles can handle PDFs, but what I just found out is that Kindles can do the Stanford Encyclopedia of Philosophy quite well.  KindleI downloaded the webpage and then deleted the content from the top of the page down to the start of the article.  Then I used Calibre to convert the webpage into Kindle format.  It looks great and the images come out well.

So I did some poking around, and found out that if you use a Firefox extension called “Download Them All” you can download a webpage and all the links on that webpage, i.e. you could go the contents page of the SEP and then download all the articles linked there.  Basically you could have the entire SEP on your Kindle and be good to go.

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Spell Sorites. No really.

Since none of the other philosophy blogs I follow have mentioned it, one of the final round contestants of the National Spelling Bee was eliminated last night by misspelling “sorites.”   I believe the contestant put a ‘p’ in front of the word.  It makes me wonder if these kids know how to do anything other than spell words.

Posted in argumentation, news, random idiocy. Tagged with , .