10.25.08

The Deal with ‘Deal or No Deal’

Posted in game theory, logic at 1:15 pm by nogre


I just saw the hit game show ‘Deal or No Deal‘.  It wasn’t the first time, but this episode had a contestant with folksiness to rival Palin, so I was entertained and kept watching.

But is there any gamesmanship to the ‘Deal or No Deal’ gameshow?  The short answer is: No.

The show begins with the contestant choosing a briefcase that contains a number that represents a real monetary amount.  The case is chosen from a group of 26 cases, with the monetary amounts ranging from a penny to a million dollars.  Recently, to up the suspense, the show has removed some of the lower amounts of money and replaced them with more million dollar cases.

The show I saw had 8 of the 26 cases carrying the million dollar value.  So when the contestant makes the initial selection, there is a slightly less than 1/3 chance of picking a million dollar case.  This case is then set aside.

The contestant then proceeds to pick other cases which are immediately opened, revealing the monetary amount they represent.  These cases are removed from the pool of cases.  After a few cases have been removed, the contestant is offered a sum of money to stop playing.  If many of the cases that have been removed were low in value, i.e. most of the million (and other high value) cases remain, then the offer will be closer to the high value cases.  If many of the high value cases have been removed, then the offer will be closer to the lower values.  Usually the value is somewhere in the middle.

These offers are made periodically when there are many cases remaining and are made after every case for the last few.  If you go all the way to the end, then you receive whatever value is in the case you initially selected.

If winning the big prize is the goal, however, all the offers are completely irrelevant.  At the outset the case the contestant chooses has a 1/3 chance of containing the big prize.  This doesn’t change throughout the game.  Let me explain why:

The rest of the cases have the same approximate ratio of million dollar values to non-million dollar values, which the contestant chooses to open randomly.  Therefore most of the time (logically speaking and whenever I watched) this ratio stays constant all the way to the end of the game.  2 cases out of the last 6 were million dollar cases in the episode I just saw.

Of course the possibility exists that the contestant will choose all of the lower value cases such that only million dollar cases remain and hence the case he or she initially chose will necessarily be a million dollar case.

However, imagine this analogous situation.  Try to pick all the cards other than Jack, Queen, King and Ace out of a shuffled deck without looking.  What will happen is that a selection of cards will be chosen irrespective of value, randomly, leaving approximately the same ratio of face cards to non-face cards remaining  (Go try it if you don’t believe me).  The chances of picking only the low values are very small.  Deal of No Deal has been on for years here in the USA and this has never happened.  The recent, and only, million dollar winner still had to choose on the last remaining case. So this part of the game has little ultimate impact upon knowing whether or not you have selected a million dollar case.

Secondly, since the cases are opened randomly during the show, no Monty Hall-like insight can be gained as to whether or not a winning case was initially selected.  Therefore the initial probability of 1/3 remains unchanged throughout the show and all the song and dance of selecting and opening the cases is a red herring (though it is top notch song and dance provided by Mr. H. Mandel and models).

This leaves the contestant in the position of deciding whether or not to accept the offer made to stop playing part way through the game without any new information.  Since the ratio of remaining monetary values remains somewhat constant, the offer made to buy the contestant out of playing should remain somewhat stable for most of the game.  It appears however, according to Wikipedia, that the initial offers are kept artificially low to build suspense, but at the end the offers are where the mathematicians say they should be.

The decision then comes down to how badly the contestant wants/ needs the money.  If the money offered to stop playing becomes large enough to significantly, to the contestant’s mind, make a big difference, he or she will likely take the money rather than take the 2/3 chance of winning significantly less.  This is what happened during the episode today: after it was made known late in the game that a sponsor was going to make a matching donation to a national charity the lady supported, she became too afraid of losing the large amount of money that was already offered, even though she said she wanted to go till the end.

In the end, the deal with ‘Deal or No Deal’ is that it is a great deal for those who get to play.  However, it is not much of a game.  The only trick is to get yourself on the show and after that how much you take home is up to luck.

 

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09.24.08

What are Quantifiers?

Posted in epistemology, game theory, logic, philosophy at 11:11 am by nogre


What are quantifiers?  Quantifiers have been thought of things that ‘range over’ a set of objects.  For example, if I say

There are people with blue eyes

this statement can be represented as (with the domain restricted to people):

∃x(Bx).

This statement says that there is at least one person with property B, blue eyes. So the ‘Ex’ is doing the work of looking at the people in the domain (all people) and picking out one with blue eyes.  Without this ‘∃x’ we would just have Bx, or x has blue eyes.

This concept of ‘ranging over’ and selecting an individual with a specific property out of the whole group works in the vast majority of applications.  However, I’ve pointed out a few instances in which it makes no sense to think of the domain as a predetermined group of objects, such as in natural language and relativistic situations.  In these cases the domain cannot be defined until something about the people involved are known, if at all; people may have a stock set of responses to questions but can also make new ones up.

So, since the problem resides with a static domain being linked to specific people, I suggest that we find a way to link quantifiers to those people.  This means that if two people are playing a logic game, each person will have their own quantifiers linked to their own domain.  The domains will be associated with the knowledge (or other relevant property) of the people playing the game.

We could index individual quantifiers to show which domain they belong to, but game theory has a mechanism for showing which player is making a move by using negation.  When a negation is reached in a logic game, it signals that it is the other player’s turn to make a move.  I suggest negation should also signal a change in domains, as to mirror the other player’s knowledge.

Using negation to switch the domain that the quantifiers reference is more realistic/ natural treatment of logic: when two people are playing a game, one may know certain things to exist that the other does not.  So using one domain is an unrealistic view of the world because it is only in special instances that two people believe the exact same objects to exist in the world.  Of course there needs to be much overlap for two people to be playing the same game, but having individual domains to represent individual intelligences makes for a more realistic model of reality.

Now that each player in a game has his or her own domain, what is the activity of the quantifier?  It still seems to be ranging over a domain, even if the domain is separate, so the problem raised above has not yet been dealt with.

Besides knowing different things, people think differently too.  The different ways people deal with situations can be described as unique strategies.  Between the strategies people have and their knowledge we have an approximate representation of a person playing a logic game.

If we now consider how quantifiers are used in logic games, whenever we encounter one we have to choose an element of the domain according to a strategy.  This strategy is a set of instructions that will yield a specified result and are separate from the domain. So quantifiers are calls to use a strategy as informed by your domain, your knowledge.  They do not ‘range over’ the domain; it is the strategies a person uses that take the domain and game (perhaps “game-state” is more accurate at this point) as inputs and returns an individual.

The main problem mentioned above can now be addressed: Instead of predetermining sets objects in domains, what we need to predetermine are the players in the game. The players may be defined by a domain of objects and strategies that will be used to play the game, but this only becomes relevant when a quantifier is reached in the game.  Specifying the players is sufficient because each brings his or her own domain and strategies to the game, so nothing is lost, and the domain and strategies do no have to be predefined because they are initially called upon within the game, not before.

I don’t expect this discussion to cause major revisions to the way people go about practicing logic, but I do hope that it provides a more natural way to think about what is going on when dealing with quantifiers and domains, especially when dealing with relativistic or natural language situations.

 

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05.29.08

Monty Hall Update

Posted in Independence Friendly logic, game theory, logic, philosophy, science at 11:41 am by nogre


I wrote out an example playing of the Monty Hall Problem in Independence Friendly Logic as a game of incomplete information and appended it to my post here.

I also left an extended comment on Dependence Logic vs. Independence Friendly Logic about some of the tribulations encountered as a non-academic trying to get my grubby little hands on obscure logic papers.

 

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04.26.08

Dependence Logic vs. Independence Friendly Logic

Posted in Independence Friendly logic, Relativity, fun, game theory, internet, logic, philosophy at 2:59 pm by nogre


I picked up Dependence Logic: A New Approach to Independence Friendly Logic by Jouko Väänänen. I figure I’ll write up a review when I am finished with the book, but there is one chief difference between Dependence Logic and Independence Friendly Logic that needs to be mentioned.

On pages 44-47 when describing the difference between Dependence Logic and Independence Friendly Logic Väänänen says,

The backslashed quantifier,

∃xn\{xi0,…,xim-1}φ,

introduced in ref. [20], with the intuitive meaning:

“there exists xn, depending only on xi0,…,xim-1, such that φ,”

…

The slashed quantifier,

∃xn/{xi0,…,xim-1}φ,

used in ref. [21] has the following intuitive meaning:

“there exists xn, independently of xi0,…,xim-1, such that φ,”

which we take to mean

“there exists xn, depending only on variables other than xi0,…,xim-1, such that φ,”

The backslashed quantifier notation is part of what Väänänen calls ‘Dependence Friendly Logic’, and is equivalent to the ‘Dependence Logic’ that the rest of the book expounds. This backslash notation makes the difference between Dependence (Friendly) Logic and Independence Friendly Logic clear by showing that the former logic takes the notion of dependence to be fundamental whereas the latter takes independence to be fundamental. Väänänen takes this to be an advantage because he says that Dependence Logic avoids making

one ha[ve] to decide whether “other variable” refers to other variables actually appearing in a formula ?, or to other variables in the domain…

However, this treatment misses an important philosophical difference between Independence Friendly Logic and Dependence Logic. Dependence Logic is fundamentally based upon Wilfrid Hodges work, ‘Compositional Semantics for a language of imperfect information’ in Logic Journal of the IGPL (5:4 1997) 539-563, in which Hodges lays out a compositional semantics for languages such as Independence Friendly Logic using sets of assignments instead of individual assignments to determine satisfaction (T or F). Väänänen infers that Independence Friendly logic is just a bit unruly when it comes to specifying variables because he is working within a system that assumes sets of assignments are a useful and unproblematic way to determine satisfaction.

However the unseen problem of using sets of assignments is that something is added by assuming the domain is a set. For example, let’s take try to define a location and take the set of all the points in the universe. However, we immediately run into relativity: All locations are defined relative to each other and the people trying to figure out where things are, i.e. There is no predetermined set of all the points in the universe. The issue is that the domain of potential assignments, the objects in the universe, may be dependent upon the person or people using them (the players of the semantic game in this case). If the domain is dependent upon the players, the set cannot be constructed until after the players have begun the game. Therefore, if we postulate that the domain is a set at the outset then the players know something about the game that they are playing, namely that it does not depend upon them because it was predetermined.

Following this line of thought it seems possible to constructed a game in which the domain {Abelard, Eloise} is such that Abelard and Eloise are the actual people playing the game and the formula is ‘Someone x lost the game by instantiating this formula’ such that whoever instantiated that formula would win the game according to the rules. But then the formula would not be satisfied, so that player would have lost, but then it would be satisfied, a paradox. It is easy enough to declare that the domain must be independent of the players, but again this signals something about the game being played to the players before the formula to be is revealed.

Lastly there is something to be said about using logic to represent natural language here too: if you consider the set of all possible responses to some question, you are not ever considering all possible responses, but all the possible responses you can think of at that time. Therefore if we are using game semantics and imperfect information to represent natural language, then it is a mistake to predetermine the domain of all possible responses separate from the people involved. Again, the domain being linked to the people involved is at odds with the domain being a predetermined set.

Long story short, there is a very good reason for not always using sets of assignments to determine satisfaction. Depending on the situation, a set may offer non-trivial information about a game or misconstrue the game being played. Independence Friendly logic makes no assumptions about the type of game being played and is therefore of greater scope than logics that are based upon Hodges work. Of course one is free to use sets of assignments to determine satisfaction and derive set-theoretic results, but the compositionality gained comes at the price of limiting the types of games that can be played.

 

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03.30.08

The Monty Hall Problem

Posted in Independence Friendly logic, game theory, logic, measurement, philosophy at 8:44 pm by nogre


The Monty Hall Problem illustrates an unusual phenomenon of changing probabilities based upon someone else’s knowledge. On the game-show Let’s Make a Deal the host, Monty Hall, asks the contestant to choose one of three possibilities - Door One, Two or Three - with one door leading to a prize and the other two leading to goats. After the contestant selects a door, another door is opened, one with a goat behind it. At this point the contestant is allowed to switch the previously selected door with the remaining (unopened) door.

Common intuition is that this choice does not present any advantage because the probability of selecting the correct door is set at 1/3 at the beginning. Each door has this 1 out of 3 chance of having a prize behind it, so changing which door you select has no effect on the outcome.

In hindsight, this intuition is wrong. If you initially selected the first goat and then switch when you get a chance, you win. If you selected the second goat and switch, you win. If you selected the prize and switch, you lose. Therefore if you switch, you win 2 out of 3, whereas if you do not switch you win only 1/3 of the time.

So what has gone horribly wrong here:

  1. Why is most everyone’s intuition faulty in this situation?
  2. How does switching doors make any difference?
  3. When did the 1/3 probability turn into a 2/3 probability?

At the beginning of the game you have a 2 out of 3 chance of losing. Likewise the game show has a 2 out of 3 chance of winning (not giving you a prize) at the beginning of the game. Both of these probabilities do not depend upon which door the prize is behind, but only upon the set-up of a prize behind only one of three doors. For instance, an outside service (not the game show) could have set everything up such that both you and the game show would be kept in the dark: there would still be 2 goats and a prize, but neither you nor the game show would know which door led to the prize.

Now imagine that it is the game show that is playing the game. The game show is trying to win by selecting a goat. From this perspective, whichever door that was chosen is good: this door has a 2 out of 3 probability of being a winner (being a goat). Therefore when given the opportunity to change (after the outside service opens a door and shows a goat), there is no reason to do so.

Of course you, the contestant, are the one making the selection, and you do not want a goat. However, if you imagined yourself in the position of the game show at the beginning, as trying to select a goat, you would reasonably assume that, just as the game show did, you were successful in choosing a goat. When given the choice to switch, now that the other goat has been removed, it seemingly makes sense to change your selection.

In this case the easiest way to view the situation is in terms of how to lose, or by considering all the possible outcomes (as mentioned above). Though this is a guess, it seems that our first blush reaction to this problem is always to view it in terms of winning and this is the reason we do not immediately recognize the benefit in switching. We start out with a 1/3 chance of winning and switching doors doesn’t immediately seem to increase this percentage.

To answer how switching doors makes a difference we need to look more closely at the doors. The door that was initially selected has a 1 out of 3 chance of being a prize, and this does not change. If you were to play many times and ignore changing doors, then you would win 33.3% of the time. At the outset the other two doors each have the exact same chance of being a winner, 1 out of 3. So the other two doors combined have a 2 out of 3 chance of containing a winning door.

Now the game show changes the number of doors available from 3 to 2, with one door guaranteed to contain a prize. If you were presented this situation without knowledge of the previous process, then you would rightly put the chance of selecting the prize at 1 out of 2, 50%.

However, you know something about the setup: The door that was initially selected had a probability of having a prize behind it set at 1 out of 3. The thing behind the other door, though, has been selected from a stacked deck: Whatever is behind the door was selected from a group of objects with a 2 out of 3 chance of containing a prize (1/3 + 1/3). You know that the odds on this door are stacked in your favor because the game show knowingly reveals the goat: In the 2/3 case in which you have previously selected a goat, the prize is behind one of the other two doors. When the game-show reveals (and removes) a goat, it guarantees that the prize is behind the last door. Therefore switching doors at the end is equivalent to combining and selecting the probability associated with the two doors not initially selected.

If the game show did not knowingly reveal the goat, you would not be able to take advantage of the stacked deck. Imagine that you select the first door and then another door is opened randomly, revealing a goat. By randomly eliminating this door (and not looking behind the unselected doors) the door that was initially selected becomes unrelated to the present choice: Only by looking behind the unselected doors does the initial selection become fixed in reference to the other doors. Since no one looked behind the doors, some bored, but not malicious, demon could have come and switched whatever was behind the selected and remaining door and neither you nor the game-show would be able to tell. Therefore switching doors when a goat is randomly revealed provides no advantage because the initial selection cannot be related to the probable location of the prize.

Only when the contestant can fix the probable locations of the prize because the location of the prize is known by the game-show, is it possible to assign interdependent probabilities on the location of the prize and the previous selection made. The odds are then tilted in the contestant’s favor by switching away from the low probability initial selection to the door that has the combination of remaining probabilities.

The logic of this needs to be represented game-theoretically with the different quantifiers representing different players of a game of incomplete information. The game would run* like this:

Domain={prize, goat, goat}

Contestant Game Show
1. - ∃x∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=x & b=y & c=z)
2. - ∃y∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=y & c=z)
3. - ∃z∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=z)
4. ∀a/x,y,z∃b∀c/x,y,z(a=g & b=g & c=p) -
5. - ∃b∀c/x,y,z(p=g & b=g & c=p)
6. ∀c/x,y,z(p=g & g=g & c=p) -
7. ∀d∀c/x,y,z(d=g & g=g & c=p) -
8. ∀c/x,y,z(g=g & g=g & c=p) -
9. (g=g & g=g & p=p) -

Line 1 is the initial setup of the prize game: the goal is for the contestant to make his or her placement of the prize and goats match the game show’s placement. Whatever is on the left side of an = will be what the contestant thinks is behind a door and what is on the right of an = will be what the game show puts behind the door, such that each = represents a door. If the formula is satisfied then the contestant will have successfully guessed the location of the prize.

Lines 2, 3 and 4 represent the results of the Game Show placing the prize and goats. Line 5 is the result of the first move of the contestant choosing where he or she thinks the prize is: the ‘a/x,y,z’ means that whatever placed in spot a has to be done independently, i.e. without knowledge, of what x or y or z is. Then the game show reveals a goat behind one of the doors not selected by the contestant. Line 7 represents the choice that is given to the contestant to switch his or her initial placement of where the prize is. Line 8 is the important step: since the contestant does not know what is behind the doors (c/x,y,z) it looks as if there is no advantage to switching. However, the contestant does know that when making a choice to reveal a goat in line 6 that at this point the game show had to know what was behind every door. This means that c is dependent upon b which was depended upon x, y, and z. With this knowledge the contestant can figure out that there is an advantage to switching because the selection of b in line 6 fixed the locations of the prize & goats and in doing so fixed the odds. Since the odds were intially stacked against the contestant, switching to the only remaining door flips the odds in the contestant’s favor, and is done so in this example. Line 9 shows that all the contestant’s choices match up with what the game show has placed behind the doors and hence she or he has won the prize.

 

*     To do a better representation would require keeping the gameshow from not placing a prize anywhere by using a line like ‘x≠y or x≠z’. For graphical brevity I left it out.

 

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12.05.07

The Logic of Biological Relativity [draft]

Posted in Independence Friendly logic, Relativity, biology, evolution, fitness, game theory, logic, measurement, science at 7:57 pm by nogre


How can we represent biological relativity in logical notation?

Organism a is adapting relative to organism b

Aab

Organism b is adapting relative to a

Aba

Organisms a and b are adapting relative to each other

Aab & Aba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are said to be adapting relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer perspective. This means we need to take into account how the observer is adapted such that the observer(s) can be compared to the organisms in question.

To remedy this problem let quantifiers range over organisms and include witnesses to identify the specific organisms in question:

For any organism x, for any organism y, there exists an organism z and there exists an organism u such that x is adapted relative to y according to organism z, and y is adapted relative to x according to organism u.

(∀x)(∀y)(∃z)(∃u)A[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that organisms x and y are selected (logically) independent of the witness organisms defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)A[xyzu]

This statement says that for any organism x, for any organism y, there exists an organism z that does not depend on y and an organism u that does not depend on x, such that organism x as witnessed by z, and organism y as witnessed by u, are adapted relative to each other.

However, though this statement gets very close to describing biological relativity, if we consider how the witnesses witness the organisms, i.e. how z witnesses the organism x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is adapted relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x adapting relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & A[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y are adapting relative to each other, as witnessed by organisms z and u (who have the equivalent adaptations respectively to x and y), and it is not necessary to predetermine what those adaptations are.

 

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11.26.07

The Logic of Relativity [draft]

Posted in Independence Friendly logic, Relativity, game theory, logic, measurement, physics, science at 2:22 pm by nogre


How can we represent relativity in logical notation?

a is moving relative to b

Mab

b is moving relative to a

Mba

a and b are moving relative to each other

Mab & Mba

This schema is unsatisfactory because it describes the situation from an indeterminate outside perspective: a and b are moving relative to each other without regard to the observer describing the situation. Relativity applies to all the perspectives in question (with special focus on any observer perspective) and hence we need a way to include the observer.

To remedy this problem let quantifiers range over perspectives and include witness individuals to identify the specific perspectives in question:

For any perspective x, for any perspective y, there exists a perspective z and there exists a perspective u such that x is moving relative to y according to witness z, and y is moving relative to x according to witness u.

(∀x)(∀y)(∃z)(∃u)M[xyzu]

Unfortunately this formulation is insufficient because witness z is logically dependent upon both x and y (as is u as well) and we want z to only witness x and u to only witness y: as both z and u are dependent upon both x and y, both x and y must be chosen before selecting z and u. This means that perspectives x and y are selected independent of the witness perspectives defeating the purpose of having those witnesses.

Getting around this difficulty is not trivial in first order logic. There is no way in first order logic to linearly order the four quantifiers such that z only depends on x and u only depends on y (Kolak & Symons p.249 [p.40 of the pdf]). (Kolak & Symons p.249 [p.40 of the pdf]). Independence Friendly logic suffices though :

(∀x)(∀y)(∃z/∀y)(∃u/∀x)M[xyzu]

This statement says that for any perspective x, for any perspective y, there exists a perspective z that does not depend on y and a perspective u that does not depend on x, such that perspective x as witnessed by z, and perspective y as witnessed by u, are moving relative to each other.

However, though this statement gets very close to describing relativity, if we consider how the witnesses witness the perspectives, how z witnesses the perspective x, there is a problem. By stating that z witnesses x and that z is independent of y, the statement ‘x is moving relative to y as witnessed by z’ is nonsense: since z is independent of y it could not be a witness to ‘x moving relative to y.’ Likewise for u.

The solution is simple enough though:

(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[x,y])

By letting x=z, making z independent of x and dependent on y, z witnesses y from the perspective of x without requiring x to be chosen before z. Likewise for u: if y=u, u is logically independent of y and u is dependent on x, then u may be chosen before y, u is dependent as a witness to the choice of x and witnesses x from the perspective of y. Perhaps more prosaically: x and y move relatively to each other, as witnessed by z and u (who have the equivalent perspectives, respectively to x and y), and it is not necessary to predetermine what those perspectives were.

A time variable rounds everything out nicely:

(∀t)(∀x)(∀y)(∃z/∀x)(∃u/∀y)((x=z) & (y=u) & M[t,x,y])

So, at time t (say now) let’s let u be your (the reader’s) perspective and z be my (the author’s) perspective. Then this statement describes our current motions as relative to each other because my perspective depends upon y, which is your perspective and your perspective depends on x, which is my perspective. Success!

——————————

 

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