Tag Archives: games

The Prospect Theory of Kahneman and Tversky

Like it or not, we are all betting individuals. But what interactions are there between the perceived and actual probabilities of things happening and the choices made for or against them? The likelihood of their occurrence, coupled with the size of the gains or losses from anticipating and acting on them, show that people are not entirely the rational agents that we think they are.

Instead of armchair introspection, careful experimental methods were used to give us these (not so) unexpected results. What is demonstrated is that deciding individuals make asymmetric choices based on their poor understanding of relative likelihoods. All sorts of biases and poor thinking on our part contribute to non-rational evaluations of how we end up choosing between alternatives.

The findings are that the near certainty of events happening is undervalued in our estimation, and the merely possible is overvalued. So those things very likely to occur have a diminished weight in our minds, and those things unlikely but possible have an increased weight. These are called the certainty effect and the possibility effect, respectively.

  • Likely Gain (Fear)
  • Likely Loss (Hope)
  • Maybe Gain (Hope)
  • Maybe Loss (Fear)

This asymmetry in valuation leads fearful individuals to accept early settlements and buy too much insurance, or hopeful individuals to buy lottery tickets and play the casino more often then they should if choosing optimally. What factors contribute to this behavior? Emotions, beliefs, and biases, probably all play a role in these perceived payoffs between dread and excitement.

In some “Dirty Harry” movie, the lead character essentially asks “do you feel lucky, punk?”, to goad another into taking a risk. In the movie “War Games”, the supercomputer more or less temptingly asks, “would you like to play a game?”, to encourage the playing of unwinnable matches. Watch out for those that know how to play the odds of hope and fear to manipulate our prospects and decisions.

Further Reading:



View at Medium.com

Daniel Kahneman / Thinking Fast and Slow



The Devolution of Trust

The Prisoner’s Dilemma is a simple game designed to show how the success or failure of cooperation between individuals can be contingent on various factors, primarily some sort of reward. Shown above is a representative payoff matrix between two players; each square shows the two choices and the two winnings for each. Each player cooperates (A or B) or cheats (A’ or B’) with the other player, so for example if A and B’ obtains (A cooperates but B cheats) then A loses 1 and B wins 3.

Each player knows all the values of the payoff matrix so it is said they have perfect information, except they don’t know what their opponent will do. If they are rational and believe their opponent to be as well, the wisest thing to do is for both to cooperate to maximize their winnings, knowing that their opponent knows that they could also cheat. If the game is played only once, however, that is clearly not the case.

If the game is iterated, things change. If each player remembers what their opponent did previously, and it is considered to be informative for what they might do next, then the player could use it to condition their decision to cooperate or cheat. Different algorithms or personalities can be considered for the players, with more or less thinking about what to do and more or less willingness to cooperate, and it is interesting to try different strategies, all the while seeing what adjustments of the payoff matrix might do to the results.

This Evolution of Trust site is a very nice lesson in some of the complications that can result for such algorithms and adjustments. On the whole, this site indicates that rationality and consideration for others can thrive, if conditions are right. In the traditional Prisoner’s Dilemma, the reward values in the payoff matrix are usually considered to be jail sentence time (so less is better), or for the site mentioned above where I’ve taken the representative matrix, monetary value (so more is better).

One thing of note in these examples is that each player doesn’t distinguish their opponent by anything other than their posteriori plays, because these players are supposed to be all part of the same group or society. But what if there is an a priori distinction that conditions their decision? So, if your opponent is a known Y, and you are a X, then you might want to raise your social credit with your other Xs by punishing a Y, even if it punishes you or even other Xs in the long run.

For example if you are a member of gang X, you wouldn’t want to cheat against another X. But cheating against a member of gang Y might raise your in-group social capital and be as important as the value of the reward. Or you might want to punish your opponent in group Y by not granting them any benefits even at the cost of your own benefit. Such distinctions are not usually part and parcel of the Prisoner’s Dilemma game, but they would add an interesting and realistic dimension to the game.

And thus lend insight into the woes of our modern political scene and culturally diverse society.

Further Reading:






[*11.24, *11.172]


The Tangram

Further Reading:





The Glass Bead Game

No permanence is ours; we are a wave
That flows to fit whatever form it finds:
Through night or day, cathedral or the cave
We pass forever, craving form that binds.

― Hermann Hesse, The Glass Bead Game

“The Glass Bead Game”, also known as “Magister Ludi”, is the last full length book by German author Hermann Hesse. His works include many thoughtful and interesting stories, detailing the main character’s personal development and spiritual growth. Hesse won the Nobel Prize in Literature in 1946, and his books saw a resurgence of popularity in the 1960’s and 1970’s in the US.

“The Glass Bead Game” is special to me because it describes, although vaguely, a fictional game that is cultivated and played in a future idyllic setting of intellectual devotion, although the larger world is certainly a post-apocalyptic one. All human knowledge is the subject of the game, and the play somehow links mathematics, music, science, cosmology, history, poetry and literature and everything else accepted as higher learning for the imagined cultural time and place.

Is the book sexist because it describes the cloistered society of the game as being restricted to boys and men because of ability, or the idea that men are less distracted from intellectual pursuits without women around? The book is either a product of its time, or perhaps of the political setting in its fictional future. Interestingly, the main character, along with three other character’s lives shown in short stories said to be written by the main character himself, are easily associated with Carl Jung’s theory of psychological types. This is the meaning of the figure above.

Several individuals and groups have tried to imagine how the actual game or any “glass bead game” (GBG) could be played, and there are scattered links on the web, many broken over time and neglect. I agree that analogy and metaphorical thinking are key points to any GBG, as well as the other pillars of attributes nicely discussed in links below.

  • Analogy
  • Connection (or Affinity)
  • Cogitation (or Contemplation, Thought)
  • Formalism (or Rules)
  • Iconicity (or Representation)
  • Syncretism (or Objectivation, of Culture or Civilization)

(Some attributes have been substituted by thesaurus for word length.)

Further Reading:

Hermann Hesse / The Glass Bead Game










… For although in a certain sense and for light-minded persons non-existent things can be more easily and irresponsibly represented in words than existing things, for the serious and conscientious historian it is just the reverse. Nothing is harder, yet nothing is more necessary, than to speak of certain things whose existence is neither demonstrable nor probable. The very fact that serious and conscientious men treat them as existing things brings them a step closer to existence and to the possibility of being born.

[*8.138, *10.82]


A Game of Fourfolds, Part 5

In this fifth installment of our ongoing series, I propose that a game could be played by making a set of equally sized and shaped triangular tiles with simple words or phrases on them. The triangles are all isosceles right triangles, also called monoboloes, so that two of them joined along their long edge would be a square, and four of them joined at their right angles would be a larger square. Figures of two tiles joined along any edge of equal length are called diaboloes, three are called triaboloes, four are called tetraboloes, and in general the figures are called polyboloes (or also polytans, after the Chinese tangram puzzle).

The words or short phrases on the monoboloes would need to be chosen judiciously so that each word has a matching opposite. (A list of such pairs of opposites or duals can be found at my previous fourfold game post.) This is so that a square diabolo could be formed from opposites, and a square tetrabolo could be formed that makes some conceptual sense. In fact, the game play would require that tiles should only be played and joined if there was a rational or explainable reason for their combination.

For example, “Water” and “Fire” could be aligned along their long edge as well as a short edge, whereas “Earth” and “Below”, not being opposites, could only be aligned along a short edge. Opposites could also be aligned “corner” to “corner” (where corner is the 90 degree angle), if there is a supporting tile between them.

During game play, the players alternate playing tiles from their hand onto the table, or pick tiles up from the table and place them back in different positions. Obviously the rules of play would need to be specified in more detail, as well as a method for scoring so that a player could “win”. Or, as a game of solitaire, perhaps winning is just maximizing the number of tiles played onto the table, or the illumination of concepts brought about by the play.

I might also consider that the flip-side of a monobolo is the same word but perhaps having white letters on a black background or colored differently to distinguish it from the “front”. And would the flip-sides all be of the same color? As I have shown various fourfolds on this blog, I have tried to orient them in a common conceptual “direction”, although that is often not clear to me or agreed upon by others of similar temperament. Perhaps they could be the same color if they metaphorically point this same way.

Also, by design and by construction, the monoboloes could be considered “Words”, diaboloes could be considered “Thoughts”, triaboloes could be considered “Actions”, and tetraboloes could be considered “Things”. This would be more in line with the hierarchy given by Richard McKeon’s 1972 lectures on Aristotle’s “Topics”. Words, thoughts, actions, and things are called “commonplaces” by McKeon, or a “place within which inquiry about meanings that are about things that are covered by that meaning can take place”.

The association of these tiles with tangrams is an interesting one. The standard tangram set consists of two small tans (unitans?), three bitans (square, midtan?, and paratan?), and two tetratans that form larger tans (bigtans?). I wonder if there is a standard nomenclature for these pieces, because mine seems rather silly.

I used to have a tangram set when I was a child and even still have an old Dover book by Ronald C. Reed “Tangrams: 330 puzzles”. It’s nice to see that it’s still available on Amazon. Of course the arrangement of the pieces in tangrams is much more flexible than what I’m proposing here for my game so really they are very little alike.

Further Reading:




Richard McKeon / Disciplines, Arts, and Faculties: Invention and Justification: Topics, Lectures given at University of Chicago 1972. (Taped, Transcribed and Edited by Patrick F. Crosby, by private communication)


Possible names for tile combinations:

  • Unit, Solitary, Unitary, Simple, Singular, Singleton
  • Binary, Duplex, Dual, Twofold, Bipartite
  • Triple, Threefold, Ternary, Trinity, Tripartite
  • Quaternary, Quadruple, Tetrad, Fourfold, Quadripartite



The Four Last Things

Over the years, there has been much religious consideration of the four last things in Christian Eschatology. They are

  • Death
  • Judgment
  • Heaven
  • Hell

Many books have been written and many paintings have been painted. In my project to present every fourth thing, I imagine I’ll come to them eventually, but why wait until then? Or maybe I’m thinking of the “Last Four Things”.

Actually, I’d much rather play the video game on my Mac, because it looks like a lot more fun. Plus it uses classical music for a background soundtrack and images from Hieronymus Bosch’s paintings, such as the Garden of Earthly Delights. When will the Mac version be available? Anyone?

Further Reading:





Robert Southwell / A Four-fold Meditation of the Four Last Things


FLT is finally out on Android and iOS! There is much rejoicing!




A Game of Fourfolds, Part 4


Many of the fourfolds presented here are roughly a combination of two dualities, a double dual if you will. Their diagrams could be considered as a crossed pair of rectangular cards, with each card showing a single pair of opposites. If one creates cards for every unique and important dual, new juxtapositions not thought of previously may be revealed by random and spontaneous association.

Of course many fourfolds cannot be reduced to the simple sum of their parts, or even the sum of their pairs. All four concepts often ramify themselves and each other due to binary, tertiary, and quaternary relations. Then the fourfold is greater than its individual constituents.

For example, the Four Elements are more than the opposite pairs of Air and Earth, Fire and Water. In Hjelmslev’s Net, Substance and Form combines with a superficially similar Content and Expression. However, above is an example of what I am striving for when Space and Time is combined with Matter and Energy.

Below is a list of dualities that might be used to create a useful set of cards. Some duals will come from fourfolds mentioned here but others will be new. Dualities or dichotomies are usually included in lists of opposites or antonyms, although they are usually more philosophical in nature.

Fourfolds that cannot be readily divided into two duals may by presented by square cards, perhaps called “trumps” or “major arcana” (or perhaps even “arcana quadra”). If cards are picked randomly but placed by choice, the rules of such actions must next be determined.

List of Duals (alphabetic):

Above, Below
Absence, Presence
Absolute, Relative
Abstract, Concrete
Active, Passive
Actual, Potential (Actual, Possible)
Addition, Subtraction
After, Before
All, None
Analytic, Synthetic
Answer, Question
And, Or
A Posteriori, A Priori
Artificial, Natural
Asymmetric, Symmetric
Atom, Void
Beautiful, Horrible
Begin, End (Start, Stop)
Being, Becoming
Belief (Faith), Doubt
Big, Little
Birth, Death
Black, White
Body, Mind
Bondage, Freedom
Bounded, Infinite
Cause, Effect
Chaos, Order (Discord, Harmony)
Child, Parent
Clean, Dirty
Combine, Separate
Complex, Simple
Content, Expression
Contingent, Necessary
Continuous, Discrete
Create, Destroy
Crooked, Straight
Dark, Light
Dawn, Dusk
Day, Night
Dead, Live
Decrease, Increase
Demand, Supply
Difference, Sameness (Distinction, Similarity)
Disease, Health (Sick, Well)
Division, Multiplication
Down, Up
Dynamic, Static
Electrical, Magnetic
Emotion, Reason (Irrational, Rational)
Empirical, Rational
Empty, Full
Enemy, Friend
Energy, Matter
Ends, Means
Even, Odd
Evil, Good
False, True
Far, Near
Fast, Slow
Female, Male
Fool, Sage
Forget, Remember
Found, Lost (Find, Lose)
Form, Substance
Future, Past
Gather, Scatter
Give, Take
Global, Local
Greater, Lesser
Guest, Host
Happiness, Sadness
Hate, Love
Hero, Villain
Hidden, Revealed (Invisible, Visible)
Higher, Lower
Holoscopic, Meroscopic
Illogic, Reason
Illusion, Reality
Immanent, Transcendent
Inside, Outside (Internal, External)
Left, Right
Listen, Speak
Long, Short
Many, One
Me, You (Them, Us)
Mix, Sort
Moon, Sun
Nature, Culture
Nature, Nurture
Negative, Positive
New, Old
Object, Subject (Objective, Subjective)
Ontic, Phenomenal
Other, Self
Part, Whole
Particle, Wave
Particular, Universal
Peace, War
Permanent, Temporary
Play, Work
Practice, Theory
Quality, Quantity
Reap, Sow
Religion, Science
Read, Write
Right, Wrong
Rough, Smooth
Private, Public (Personal, Social)
Profane, Sacred (Secular, Spiritual)
Pull, Push
Space, Time
Strong, Weak
Vice, Virtue

List of Trumps:

Air, Earth, Fire, Water
Cold, Hot, Dry, Wet
East, West, North, South
Fall, Winter, Spring, Summer
Dawn, Day, Dusk, Night



Vocabulary list by Opposites (or Antonyms)




[*9.34, *9.37]



Snakes and Ladders

My propositions are elucidatory in this way: he who understands me finally recognizes them as senseless, when he has climbed out through them, on them, over them. (He must so to speak throw away the ladder, after he has climbed up on it.) He must surmount these propositions; then he sees the world rightly.

— Ludwig Wittgenstein, from Tractatus Logico-Philosophicus 6.54

Consider the children’s game of “Snakes and Ladders”.

The object of the game is to traverse a sequence of locations on a game board, numbered from one (the beginning) to some maximum goal (the end), with the numbers increasing from the bottom of the board to the top. The player usually competes with other players to win the goal first, because otherwise why do it. Ladders join lower numbers at their base with higher numbers at their top. Snakes join higher numbers at their head with lower numbers at their tail.

The players take turns and each move forward a number of locations determined with a roll of a die. If a player lands on the bottom of a ladder, she can immediately climb to the top of it, skipping the locations in between. However if she lands on the head of a snake, she must slide down the snake to its tail, essentially losing her recent gains of several turns.

The game progresses by chance, and indeed the winner is completely chosen by luck. There are no choices that players can make to increase their chance of winning the game, and so it is completely random. It is evidently an ancient game and older versions often portrayed the ladders as virtues and the snakes as vices. What can this game teach us about morality and life in general?

Unfortunately the player cannot gain knowledge or experience along their journey to help them. There is no way to increase one’s chance to land on a ladder’s bottom or decrease one’s chance to land on a snake’s head. The game is on the other end of the spectrum from a choice driven game such as tic-tac-toe, or even chess. I’m also reminded of the card game “War”, which is another random game played by children. Such completely random games equalize chances of winning among different ages, I imagine.

From Wikipedia: The game (Snakes and Ladders) is a central metaphor of Salman Rushdie’s Midnight’s Children. The narrator describes the game as follows:

All games have morals; and the game of Snakes and Ladders captures, as no other activity can hope to do, the eternal truth that for every ladder you hope to climb, a snake is waiting just around the corner, and for every snake a ladder will compensate. But it’s more than that; no mere carrot-and-stick affair; because implicit in the game is unchanging twoness of things, the duality of up against down, good against evil; the solid rationality of ladders balances the occult sinuosities of the serpent; in the opposition of staircase and cobra we can see, metaphorically, all conceivable oppositions, Alpha against Omega, father against mother.


Actually, it reveals the fourness of things. The snake tails and ladder tops are different in kind from the snake heads and ladder bases, because landing on the former the player doesn’t change position. She can instead contemplate possibly backsliding and revisiting this location via snake or the experiences of having taken the long way instead of just climbing to this location via  ladder.

A more modern version could replace the snakes and ladders with one entity, perhaps wormholes, which would deliver one instantly to the opposite end, either forward or backward. The name could be revised to “Shortcuts and Backtracks”, perhaps. Or maybe it would be too confusing.




[*8.144, *9.102]


A Game of Four-folds, Part 3


Here’s another idea for the Game of Fourfolds, instead of using hexaflexagons. Since each fourfold can be permuted six ways, how about arranging those in six squares on the faces of a cube?

Here are the permutations of Heidegger’s “das Geviert” arranged on the faces of a cube, if one cuts it out and folds it up properly. Not only is the square the regular polygon of materiality, the cube is the regular polyhedron of Earth, the most material of the ancient four elements.

It might be awesome to have a cube of every fourfold I’ve talked about (at least the good ones), to combine and compare them. Even more awesome are the cubes for the REAL elements I’ve seen when I’ve searched images for “element cube”.

What I want to know is whether the six permutations can be arranged in a symmetric way on a cube?



A Game of Four-folds, Part 2


One way to have all six arrangements of the quadrants for each four-fold card is to make a tetraflexagon for each card. Above are the two sides of a sheet that should be printed out by flipping it on the short edge, so that the upper right corner of the bottom image will end up behind the upper left corner of the top image. Then fold the tetraflexagon by the simple instructions found at the link below. Then you will have all six arrangements of the four-fold for the Four Elements! There are actually seven states for the tetraflexagon, six of which have a valid arrangement on just one side of it, and one that doesn’t have a valid arrangement on either side. Seems like a lot of work, though.

How to Fold a Hexa-tetraflexagon