We recall that a *periodic point* of *period* for a function is a point such that . With this definition, a periodic point of period is also periodic of period for every which is a multiple of . If but for every from 1 to , we say that is the *least period* of .

**Theorem 1. (Sharkovsky’s “little” theorem)** *Let be an interval and let be a continuous function su. If has a point of least period 3, then it has points of arbitrary least period; in particular, it has a fixed point.*

Note that no hypothesis is made on being open or closed, bounded or unbounded.

Our proof of Sharkovsky’s “little” theorem follows the one given in (Sternberg, 2010), and could even be given in a Calculus 1 course: the most advanced result will be the intermediate value theorem.

**Lemma 1.** *Let be a compact interval of the real line, let be a continuous function. Suppose that for some compact interval it is . Then has a fixed point in .*

*Proof.* Let and be the minimum and the maximum of in , respectively. As , it is and . Choose such that and . Then is nonpositive at and nonnegative at . By the intermediate value theorem applied to , must have a fixed point in the closed and bounded interval (possibly reduced to a single point) delimited by and , which is a subset of .

**Lemma 2.** *In the hypotheses of Lemma 1, let be a closed and bounded interval contained in . Then there exists a closed and bounded subinterval of such that .*

Proof. Let . We may suppose , otherwise the statement is trivial. Let be the largest such that . Two cases are possible.

- There exists such that . Let be the
*smallest*such , and let . Then surely , but if for some we had either or , then by the intermediate value theorem, for some we would also have either or , against our choice of and . - for every . Let then be the largest such that , and let . Then for reasons similar to those of the previous point.

*Proof of Sharkovsky’s “little” theorem.* Let be such that , , and . Up to cycling between these three values and replacing with , we may suppose . Fix a positive integer : we will prove that there exists such that and for every .

Let and be the “left” and “right” side of the closed and bounded interval : then and by the intermediate value theorem. In particular, , and Lemma 1 immediately tells us that has a fixed point in . Also, , so also has a point of period 2 in , again by Lemma 1: call it . This point cannot be a fixed point, because then it would also belong to as , but which has period 3. As we can obviously take , we only need to consider the case .

By Lemma 2, there exists a closed and bounded subinterval of such that . In turn, as , there also exists a closed and bounded subinterval of such that , again by Lemma 2: but then, . By iterating the procedure, we find a sequence of closed and bounded intervals such that, for every , and .

We stop at and recall that : we are still in the situation of Lemma 2, with in the role of . So we choose as a closed and bounded subinterval not of , but of , such that . In turn, as , there exists a closed and bounded subinterval of such that . Following the chain of inclusions we obtain . By Lemma 1, has a fixed point in , which is a periodic point of period for .

Can the least period of for be smaller than ? No, it cannot, for the following reason. If has period , then so has , and in addition is divisible by . But while for every : consequently, if has period , then . But this is impossible, because by construction as , while .

Theorem 1 is a special case of a much more general, and complex, result also due to Sharkovsky. Before stating it, we need to define a special ordering on positive integers.

**Definition.** The *Sharkovsky ordering* between positive integers is defined as follows:

- Identify the number , with odd integer, with the pair .
- Sort the pairs with in lexicographic order.

That is: first, list all the odd numbers, in increasing order; then, all the doubles of the odd numbers, in increasing order; then, all the quadruples of the odd numbers, in increasing order; and so on.

For example, and - Set for every and .

That is: the powers of 2 follow, in the Sharkovskii ordering, any number which has an odd factor.

For example, . - Sort the pairs of the form —i.e., the powers of 2—in reverse order.

The set of positive integer with the Sharkowsky ordering has then the form:

Note that is a total ordering.

**Theorem 2. (Sharkovsky’s “great” theorem)** *Let be an interval on the real line and let be a continuous function.*

*If has a point of least period , and , then has a point of least period . In particular, if has a periodic point, then it has a fixed point.**For every integer it is possible to choose and so that has a point of minimum period and no points of minimum period for any . In particular, there are functions whose only periodic points are fixed.*

Bibliography:

- Keith Burns and Boris Hasselblatt. The Sharkovsky theorem: A natural direct proof.
*The American Mathematical Monthly***118(3)**(2011), 229–244. doi:10.4169/amer.math.monthly.118.03.229 - Robert L. Devaney,
*An Introduction to Chaotic Dynamical Systems*, Second Edition, Westview Press 2003. - Shlomo Sternberg,
*Dynamical Systems*, Dover 2010.

Find the talk on my personal blog HERE

]]>Let’s start from the beginning:

*Definition 1.* Let be a semigroup. A function is *subadditive* if:

for every (1)

If is a group, with an identity element and a multiplicative inverse for every element , then (1) is equivalent to:

for every

Usually, we will have be one of the sets and of integers and reals, respectively, or one of the sets and of positive integers and positive reals, respectively. All these sets will be considered as semigroups with respect to addition.

Examples of subadditive functions are:

- The
*Heaviside function*defined by if and if . This function is subadditive, because if and are both negative, then the left-hand side of (1) is 0, and if one of them is nonnegative, then the right-hand side is either 1 or 2. - Let and let be defined by if and if . Then is subadditive, because the left-hand side of (1) is either 1 or 2, and the right-hand side is either 2, 3, or 4. For and this shows that a subadditive function can be discontinuous at every point.
- Let be a finite nonempty set and let be the
*free monoid*on , that is, the set of words on with concatenation as the binary operation and the*empty word*as the identity element. The length of a word is a subadditive (actually, additive) function on .

If is a subsemigroup of either or , a useful trick is that is subadditive on if and only if is subadditive on . This, for example, allows to “dualize” proofs on to make them work on , the additive semigroup of negative reals.

**Fekete’s lemma.** Let be a subadditive function.

- If or , then .
- Dually, if or , then .
- Finally, if or , then , and both are finite.

Note that cannot be , but can be : for example, is subadditive on and . Dually, cannot be , but can be . Note that itself needs not be subadditive: for example, is subadditive on , but is not.

*Proof of point 1 with :* Fix a positive integer . Every positive integer large enough can be written in the form with positive integer and (attention!) . By subadditivity,

.

But by construction, : since there are no more than possible values for , by taking the upper limits we get

.

This holds for every positive integer , so we can conclude:

.

A key ingredient of the proof is that is bounded on . The argument can be modified to work on , but requires that be bounded in every closed and bounded interval of the form : this is actually true if is subadditive, but proving this fact would go beyond the scope of our talks.

An immediate consequence of Fekete’s lemma is that, as it was intuitively true from the definition, a subadditive function defined on or can go to for at most linearly. On the other hand, an everywhere negative subadditive function defined on positive reals or positive integers can go to for arbitrarily fast. Indeed, the following holds:

**Lemma 1.** Let be either or and let . If is negative and subadditive and is positive and nondecreasing, then is subadditive (and negative).

*Proof:* If and , then . Then the following chain of inequalities hold:

.

Hence, if and is positive and nondecreasing, then is subadditive and .

To see an application of Fekete’s lemma in the context of the theory of dynamical system, we introduce the notions of subshift and of cellular automaton. We will first do so in dimension 1, then expand to arbitrary dimension in later talks.

*Definition 2.* Let be a finite nonempty set. A subset of the set of bi-infinite words is a (one-dimensional) *subshift* if there exists a set of *forbidden words* such that the following holds: for every , it is if and only if for no and it is . The set of the words over which appear in elements of is called the *language* of the subshift .

Examples of subshifts are:

- The
*full shift*. In this case, . - The
*golden mean shift*on the binary alphabet, where .

Let be a subshift on and let and be words on of length and , respectively. If is an allowed word for , then so must be and : that is, there cannot be more allowed words of length than juxtapositions of an allowed word of length and an allowed word of length . Switching to logarithms, we see that

is a subadditive function of the positive integer variable : Fekete’s lemma then tells us that

(2)

exists. The quantity (2) is called the *entropy* of the subshift , and is a measure of the quantity of information it can convey.

(As a funny note, the use of the uncapitalized letter to indicate entropy apparently originates from Claude Shannon, after John von Neumann suggested that he called “entropy” a similar information-theoretical quantity. Shannon decided to uncapitalize the letter used by Ludwig Boltzmann… which, however, was a capitalized .)

*Definition 3.* A one-dimensional *cellular automaton* is a triple where is a finite nonempty *set of states*, is a nonnegative integer *radius*, and is a *local update rule*.

A cellular automaton induces a *global transition function* by synchronous update:

for every (3)

If is the global transition function of a cellular automaton with set of states , then is a subshift. Not every subshift can be obtained this way: those that can, belong to the special class of *sofic shifts*, a term suggested by Benjamin Weiss coming from the Hebrew word for “finite”.

In the upcoming talk (or talks) we will examine the case of several variables, and correspondingly, subshifts and cellular automata in higher dimension. In particular, we will discuss a generalization of Fekete’s lemma to arbitrarily many positive integer variables.

Bibliography:

- Silvio Capobianco. Multidimensional cellular automata and generalization of Fekete’s lemma. Discrete Mathematics and Theoretical Computer Science 10 (2008), 95–104.
- Einar Hille. Functional Analysis and Semigroups. American Mathematical Society, 1948.
- Douglas Lind and Brian Marcus. An Introduction to Symbolic Dynamics and Coding. Cambridge University Press 1995.
- Tommaso Toffoli, Silvio Capobianco, and Patrizia Mentrasti. When—and how—can a cellular automaton be rewritten as a lattice gas? Theoretical Computer Science 403 (2008), 71–88.

First of all, let us recall what the Church-Turing thesis is, and what it is not. Its statement, as reported by the Stanford Encyclopedia of Philosophy, goes as follows:

A function of positive integers is effectively calculable only if recursive.

Here, for a calculation procedure to be “effective” means the following:

- it has a finite description;
- it always returns the correct output, given any valid input;
- it can be “carried on by pencil and paper” by a human being; and
- it requires no insight or ingenuity on the human’s behalf.

One model of effective procedures is given by the recursive functions; another one, by the functions computable by Turing machines; a third one, by the functions which are representable in Church’s -calculus. Alan Turing and Stephen Cole Kleene proved that the three classes coincide: thus, in ordinary practice, the Church-Turing thesis is often stated with “Turing-computable” in place of “recursive”.

The class of Turing machines has the advantage of containing a *universal element*: a special Turing machine and an encoding from the set of Turing machines to the set of natural numbers exists such that, when the special Turing machine is provided the encoding of an arbitrary Turing machine and a valid input for the latter, it will return the value of the encoded Turing machine on the provided input.

Now that we have written down what the Church-Turing thesis is, we can examine Akl’s theorem.

In his 2005 paper, Akl defines a universal computer as a system having the following features:

- It has means of communicating with the outside world, so to receive input, and where to send its output.
- It can perform every elementary arithmetic and logic operations.
- It can be programmed, according to the two previous rules.
- It has unlimited memory to use for input, output, and temporary values.
- It can only execute finitely many operations (evaluating input, producing output, performing an elementary operation, etc.) at each time step.
- It can simulate any computation performed by any other model of computation.

The statement of the theorem, which does not appear explicitly in the original paper but is written down in the one from 2015 which clarifies the idea and addresses criticism, is hereby reported *verbatim*:

**Nonuniversality in Computation Theorem (NCT):** No computer is universal if it is capable of exactly operations during time unit of computation, where is a positive integer, and is finite and fixed once and for all.

The main argument is that no such computer can perform a computation which requires more than operations at some time . Explicit examples happen in parallel computation, a field Akl is a master of, where the number of operations that can be performed in a time unit grows linearly with the number of processors: for instance, reading values in input can be done in time by a parallel machine with processors, but not by any machine with processors.

Such requirement, however, does not appear in the notion of universality at the base of the original, and actual, Church-Turing thesis. There, to “simulate” a machine or algorithm means to be able of always reproducing the *same output* of the algorithm, given *any valid input* for it, *up to an encoding* of the input and the output. But no hypothesis on *how* the output is achieved from the input is made: a simulation in *linear time*, such that each step of the simulated algorithm is reproduced by exactly operations of the Turing machine, is as good as one where the simulation of the th step takes operations from the Turing machine, or where no such regularity appears.

Among the (counter)examples provided by Akl are:

- Computations with time-varying variables.
- Computations with time-varying computational complexity.
- Computations whose complexity depends on their placement on a schedule.
- Computations with interacting variables, e.g., states of entangled electrons.
- Computations with uncertain time constraints.

None of these, however, respect the definition of computation from the model of recursive functions: where the values of the variables are given once and for all, and can possibly change for recursive calls, but not for the original call. They can be seen as instances of *unconventional* models of computation: but by doing this, one changes the very notion of computation, which ceases to be the one at the basis of the Church-Turing thesis.

So my guess is that Akl’s statement about the falsity of the Church-Turing thesis actually falls in the following category, as reported in the humorous list by Dana Angluin:

*Proof by semantic shift:* Some standard but inconvenient definitions are changed for the statement of the result.

Actually, if we go back to Akl’s definition of a universal computer, it appears to be fine until the very last: the first two points agree with the definition of effective computation at the basis of the actual Church-Turing thesis, the next three are features of any universal Turing machine. The problem comes from the last point, which has at least two weak spots: the first one being that it does not define precisely what a model of computation is, which can be accepted as Akl is talking of unconventional computation, and it is wiser to be open to other possibilities. But there is a more serious one, in that it is not clear

*what does the expression “to simulate” mean*.

Note that the Stanford Encyclopedia of Philosophy reports the following variant of the Church-Turing thesis, attributed to David Deutsch:

Every finitely realizable physical system can be perfectly simulated by a universal model computing machine operating by finite means.

Deutsch’s thesis, however, does not coincide with the Church-Turing thesis! (This, notwithstanding Deutsch’s statement that “[t]his formulation is both better defined and more physical than Turing’s own way of expressing it”.) Plus, there is another serious ambiguity, which is of the same kind as the one in Akl’s definition:

*what is “perfectly simulated” supposed to mean?*

Does it mean that every single step performed by the system can be reproduced in real time? In this case, Akl is perfectly right in disproving it *under the constraint of boundedly many operations at each time unit*. Or does it mean that the simulation of each elementary step of the process (e.g., one performed in a quantum of time) ends with the correct result if the correct initial conditions are given? In this case, the requirement to reproduce exactly what happens between the reading of the input and the writing of the output is null and void.

Worse still, there is a vulgarized form of the Church-Turing thesis, which is reported by Akl himself on page 172 of his 2005 paper!, and goes as follows:

Any computable function can be computed on a Turing machine.

If one calls *that* “the Church-Turing thesis”, then Akl’s NCT is absolutely correct in disproving it. But *that* is not the actual Church-Turing thesis! It is actually a rewording of what in the Stanford Encyclopedia of Philosophy is called “Thesis M”, and explicitly stated not to be equivalent to the original Church-Turing thesis—and also false. Again, the careful reader will have noticed that, in the statement above, being “computable by a Turing machine” is a well defined property, but “computable” *tout court* definitely not so.

At the end of this discussion, my thesis is that Akl’s proof is correct, but NCT’s consequences and interpretation might not be what Akl means, or (inclusive disjunction) what his critics understand. As for my personal interpretation of NCT, here it goes:

*No computer which is able to perform a predefinite, finite number of operations at each finite time step, is universal across all the different models of computation, where the word “computation” may be taken in a different meaning than that of the Church-Turing thesis.*

Is mine an interpretation by semantic shift? Discussion is welcome.

References:

- Selim G. Akl. The Myth of Universal Computation. Parallel Numerics ’05, 167–192.
- Selim G. Akl. Nonuniversality explained.
*International Journal of Parallel, Emergent and Distributed Systems***31:3**, 201–219. doi:10.1080/17445760.2015.1079321 - The Church-Turing Thesis. Stanford Encyclopedia of Philosophy. First published January 8, 1997; substantive revision August 19, 2002. http://plato.stanford.edu/entries/church-turing/
- Dana Angluin’s List of Proof Techniques. http://www.cs.northwestern.edu/~riesbeck/proofs.html

We consider languages made of symbols that represents either objects, or functions, or relations: in particular, unary relations, or equivalently, sets. A *sentence* on such a language is a finite sequence of symbols from the language and from the standard logical connectives and quantifiers ( for conjunction, for disjunction, for negation, etc.) according to the usual rules, such that every variable is bounded by some quantifier. A *first-order* sentence only has quantifiers on objects, while a *second-order* sentence can have quantifiers on functions and relations (in particular, sets) as well.

For example, the set is made of the following first-order sentences on the language :

Of course, second-order logic is much more expressive than first order logic. The natural question is: *how much*?

The answer is: possibly, *too much* more than we would like.

To discuss how it is so, we recall the notion of *model*. Informally, a model of a set of sentences is a “world” where all the sentences in the set are true. For instance, the set of natural numbers with the usual zero, successor, addition, multiplication, and ordering is a model of . A model for a set of sentences is also a model for every *theorem* of that set, *i.e.*, every sentence that can be derived in finitely many steps from those of the given set by applying the standard rules of logic.

For sets of first-order sentences, the following four results are standard:

**Compactness theorem.** (Tarski and Mal’tsev) Given a set of first-order sentences, if every finite subset of has a model, then has a model.

**Upwards Löwenheim-Skolem theorem.** If a set of first-order sentences has a model of infinite cardinality , then it also has models of every cardinality .

**Downwards Löwenheim-Skolem theorem.** If a set of first-order sentences on a finite or countable language has a model, then it also has a finite or countable model.

**Completeness theorem.** (Gödel) Given a set of first-order sentences, if a first-order sentence is true in every model of , then is a theorem of .

All of these facts fail for second-order theories. Let us see how:

We start by considering the following second-order sentence:

**Lemma 1.** The sentence is true in a model if and only if the universe of is at most countable.

The informal reason is that intuitively means:

the universe is a monoid on a single generator

Let us now consider the following second-order sentence:

**Lemma 2.** The sentence is true in a model if and only if the universe of is infinite.

The informal reason is that intuitively means:

the universe contains a copy of the natural numbers

**Theorem 1.** Both Löwenheim-Skolem theorems fail for sets of second-order sentences.

*Proof.* only has countably infinite models. only has uncountably infinite models.

Let us now consider the set of all the sentences of together with the following second-order sentence:

Clearly, is the induction principle: which is an axiom in second-order Peano arithmetics, but only an *axiom scheme* in first-order PA.

**Lemma 3.** Every model of is isomorphic to the set of natural numbers with zero, successor, addition, multiplication, and ordering.

The informal reason is that , though finite, is powerful enough to tell numbers from each other: therefore, in every model of , each numeral (th iteration of the successor, starting from ) can be denoted by at most one item in the universe of the model. On the other hand, is powerful enough to reconstruct every numeral.

**Theorem 2.** The compactness theorem fails for sets of second-order sentences.

*Proof.* Let be a constant outside the language of . Consider the set made of all the sentences from and all the sentences of the form . Then every finite subset of has a model, which can be obtained from the set of natural numbers by interpreting as some number strictly greater than all of the values such that . However, a model of is also a model of , and must be isomorphic to the set of natural numbers: but no interpretation of the constant is possible within such model.

We can now prove

**Theorem 3.** The completeness theorem does not hold for second-order sentences.

In other words, *second-order logic is semantically inadequate*: it is not true anymore that all “inequivocably true” sentences are theorems. The proof will be based on the following two facts:

**Fact 1.** (Gödel) The set of the first-order formulas which are true in *every* model of is recursively enumerable.

**Fact 2.** (Tarski) The set of first-order formulas which are true in *is not* recursively enumerable.

Fact 1 is actually a consequence of the completeness theorem: the set of first-order formulas which are true in every model of is the same as the set of first-order sentences that are provable from , and that set is recursively enumerable by producing every possible proof! To prove Theorem 3 it will thus be sufficient to prove that Fact 1 does not hold for second-order sentences.

*Proof of Theorem 3.* We identify with the conjunction of all its formulas, which are finitely many.

Let be a first-order sentence in the language of . Because of what we saw while discussing the compactness theorem, is true in if and only if it is true in every model of : this, in turn, is the same as saying that is true in every model of . Indeed, let be a model of : if is isomorphic to , then is true in if and only if is true in ; if is not isomorphic to , then is false in , which makes true in . This holds whatever is.

Fix a Gödel numbering for sentences. There exists a recursive function that, for every sentence , transforms the Gödel number of the first-order sentence into the Gödel number of the second-order sentence .

Suppose now, for the sake of contradiction, that the set of second-order sentences that are true in every model of is recursively enumerable. Then we could get a recursive enumeration of the set of first-order sentences which are true in the standard model of by taking the Gödel number of such a sentence , turning it into that of via the aforementioned recursive function, and feeding the latter number to the semialgorithm for second-order sentences that are true in every model of . But because of Tarski’s result, no such recursive enumeration exists.

Bibliography:

- George S. Boolos et al. Computability and Logic. Fifth Edition. Cambridge University Press, 2007

Given a base , consider the base- writing of the nonnegative integer

where each is an integer between and . *The Cantor base-* writing of is obtained by iteratively applying the base- writing to the exponents as well, until the only values appearing are integers between and . For example, for and , we have

and also

Given a nonnegative integer , consider the *Goodstein sequence* defined for by putting , and by constructing from as follows:

- Take the Cantor base- representation of .
- Convert each into , getting a new number.
- If the value obtained at the previous point is positive, then subtract from it.

(This is called the*woodworm’s trick*.)

**Goodstein’s theorem.** Whatever the initial value , the Goodstein sequence ultimately reaches the value in finitely many steps.

Goodstein’s proof relies on the use of ordinal arithmetic. Recall the definition: an ordinal number is an equivalence class of well-ordered sets modulo *order isomorphisms*, *i.e.*, order-preserving bijections.Observe that such order isomorphism between well-ordered sets, if it exists, is unique: if and are well-ordered sets, and are two distinct order isomorphisms, then either or has a minimum , which cannot correspond to any element of .

An interval in a well-ordered set is a subset of the form .

**Fact 1.** Given any two well-ordered sets, either they are order-isomorphic, or one of them is order-isomorphic to an initial interval of the other.

In particular, every ordinal is order-isomorphic to the interval .

All ordinal numbers can be obtained via von Neumann’s classification:

- The zero ordinal is , which is trivially well-ordered as it has no nonempty subsets.
- A successor ordinal is an ordinal of the form , with every object in being smaller than in .

For instance, can be seen as . - A limit ordinal is a nonzero ordinal which is not a successor. Such ordinal must be the least upper bound of the collection of all the ordinals below it.

For instance, the smallest transfinite ordinal is the limit of the collection of the finite ordinals.

Observe that, with this convention, each ordinal is an element of every ordinal strictly greater than itself.

**Fact 2.** Every set of ordinal numbers is well-ordered with respect to the relation: if and only if .

Operations between ordinal numbers are defined as follows: (up to order isomorphisms)

- is a copy of followed by a copy of , with every object in being strictly smaller than any object in .

If and are finite ordinals, then has the intuitive meaning. On the other hand, , as a copy of followed by a copy of is order-isomorphic to : but is strictly larger than , as the latter is an initial interval of the former. - is a stack of copies of , with each object in each layer being strictly smaller than any object of any layer above.

If and are finite ordinals, then has the intuitive meaning. On the other hand, is a stack of copies of , which is order-isomorphic to : but is a stack of copies of , which is order-isomorphic to . - is if , if is the successor of , and the least upper bound of the ordinals of the form with if is a limit ordinal.

If and are finite ordinals, then has the intuitive meaning. On the other hand, is the least upper bound of all the ordinals of the form where is a finite ordinal, which is precisely : but .

*Proof of Goodstein’s theorem:* To each integer value we associate an ordinal number by replacing each (which, let’s not forget, is the base is written in) with . For example, if , then

and (which, incidentally, equals ) so that

We notice that, in our example, , but : why is it so?, and is it just a case, or is there a rule behind this?

At each step where , consider the writing . Three cases are possible:

- .

Then , as , and . - and .

Then for a transfinite ordinal , and . - and .

Then for some , and is a number whose th digit in base is zero: correspondingly, the rightmost term in will be replaced by a smaller ordinal in .

It is then clear that the sequence is strictly decreasing. But the collection of all ordinals not larger than is a well-ordered set, and *every nonincreasing sequence in a well-ordered set is ultimately constant*: hence, there must be a value such that . But the only way it can be so is when : in turn, the only option for to be zero, is that is zero as well. This proves the theorem.

So why is it that Goodstein’s theorem is not provable in the first order Peano arithmetics? The intuitive reason, is that the exponentiations can be arbitrarily many, which requires having available all the ordinals up to

, times , times:

this, however, is impossible if induction only allows finitely many steps, as it is the case for first-order Peano arithmetics. A full discussion of a counterexample, however, would greatly exceed the scope of this post.

]]>Let us recall the basic notions. In a *game in normal form* we have:

- A set of
*players*. - A set of
*strategies*for each player. - A collection of
*utility functions*which associate to each*strategic profile*a real number, such that is the*utility*player gets from the strategic profile .

A *Nash equilibrium* for a game in normal form is a strategic profile such that, for every player and every strategy feasible for player , it is the case that . We had seen that not every finite game in normal form admits a pure strategy Nash equilibrium: so, we introduced randomization.

A *mixed strategy* for player is a probability distribution . If is finite, this is the same as assigning values for . A *mixed strategy profile* is a collection of mixed strategies for each player. A *mixed strategy Nash equilibrium* is a mixed strategy profile such that, for every player and every mixed strategy feasible for player ,

.

The idea behind Nash’s proof goes as follows. If the game is finite, then a mixed strategy for player is identified with a point of

therefore, mixed strategy profiles can be identified with points of

which is compact and convex as all of its components are. Mixed strategy Nash equilibria are those points of where each pure strategy , , , is used in the most efficient way: by relaxing the condition and allowing a small “slack” with respect to such most efficient way, it is possible to define a continuous transformation of mixed strategy profiles into mixed strategy profiles, which will have a fixed point because of the Brouwer fixed-point theorem. By gradually reducing the slack, a mixed strategy Nash equilibrium is found as a limit point of such approximations.

Suppose player has available the pure strategies for . Let be an arbitrary mixed strategy profile and be an arbitrary integer. Consider the following quantities:

- .
- .
- .
- .

Given , the sum is bounded from below by , hence the functions

are continuous and nonnegative and satisfy whatever and are. As a consequence, the functions

that is,

are continuous transformations of into itself. Let be a fixed point of , whose existence is ensured by the Brouwer fixed-point theorem: as is compact, the sequence has a limit point .

Suppose, for the sake of contradiction, that is not a mixed strategy Nash equilibrium. Then there must be a player and a mixed strategy such that . The only way this may happen, is that some *pure* strategy is used *suboptimally* by , that is,

Choose and so that:

- belongs to a subsequence converging to .
- .
- .
- .

Points 2 and 3 tells us that is strictly smaller than : this, together with point 4, yields , thus . But is a fixed point for , so

:

and as may be taken arbitrarily large and be made arbitrarily small, we must conclude that too. This is a contradiction.

]]>To introduce this idea, together with other basic game-theoretic notions, we resort to some examples. Here goes the first one:

Alice and Bob are planning an evening at the cinema. Alice would like to watch the romantic movie, while Bob would like to watch the action movie. Neither of them likes much the other’s favored movie: however, should they split, the sadness for being alone would be so big, that neither of them would enjoy his or her movie!

This is the kind of situation modeled by a *game in normal form*, where we have:

- A set of
*players*. - A set of
*strategies*for each player. - A collection of
*utility functions*which associate to each*strategic profile*a real number, such that is the*utility*player gets from the strategic profile .

In the case of Alice and Bob, this may be summarized with a table such as the following:

Romantic | Action | |

Romantic | ||

Action |

Such tables represent games in normal form between two players, where the *rows* of the table are labeled with the strategies suitable for the *first* player, and the *columns* of the table are labeled with the strategies suitable for the *second* player: the entries of the table indicate the values of the utility functions when the first player plays the corresponding row and the second player plays the corresponding column. When we want to emphasize the role of player in contrast to the others, we write as , and talk about the strategy of player *given* the strategic profile of the other players.

Suppose that Alice is the first player, and Bob is the second player: then the table tells us that, if they both choose the romantic movie, Alice will enjoy it a lot (utility value ) and Bob not very much (utility value ). However, if Bob *defects* from this strategic profile and goes watch the action movie, he will ultimately not enjoy it, because he will be sad for not being together with Alice—which was the entire point about organizing the evening at the movies!

Let us consider another game (a rather serious one indeed) where the players are a lion and a gazelle. The lion wants to catch the gazelle; the gazelle wants to avoid being caught by the lion. To do this, they may choose between being on the move, or staying more or less in the same place. It turns out, from observation in the field, that the table for the lion-and-gazelle situation is similar to the one below:

Move | Stay | |

Move | ||

Stay |

We observe that, for the lion, the most profitable strategy is to move. Indeed, if the gazelle moves, then the utility for the lion is if he moves, which is more than the he gets if he stays; on the other hand, if the gazelle stays, then the utility for the lion is if he moves, which is more than the he gets if he stays. A strategy such as this, which always gives the *best* possible result *independently* of the other players’ strategies, is called a *dominant strategy*. Such strategies are indeed quite rare: indeed, neither Alice nor Bob from the previous game had a dominant strategy, nor has the gazelle here, as they can maximize their own profit only by choosing the *same* strategy as the other player.

So, what if we relax the requirement, and just demand that every player chooses the most favorable strategy, *given* the strategies of the other players? This is the basic intuition under the concept of Nash equilibrium, formalized and studied by John Nash in his 1950 doctoral thesis.

**Definition 1.** A *Nash equilibrium* for a game in normal form is a strategic profile such that, for every player and every strategy feasible for player , it is the case that .

The situation when both the lion and the gazelle are on the move, is a Nash equilibrium: and is the only Nash equilibrium in the corresponding game. (By definition, every dominant strategy enters every Nash equilibrium.) The situation when both Alice and Bob go watch the romantic movie, is a Nash equilibrium: and so is the one when they go watch the action movie.

So, does every game have a Nash equilibrium?

Actually, no.

Indeed, suppose that the predator and the prey, instead of being large mammals such as the lion and the gazelle, are small insects such as a dragonfly and a mosquito. It then turns out, after careful observation, that the table for the predator-prey game gets more similar to the following:

Move | Stay | |

Move | ||

Stay |

In this situation, the dragonfly maximizes its utility if it does the *same* as the mosquito. In turn, however, the mosquito maximizes its own utility if it does the *opposite* than the dragonfly! In such a situation there can be no such thing as a Nash equilibrium as defined above.

Where determinism fails, however, randomization may help.

**Definition 2.** A *mixed strategy* for the player in a game in normal form is a probability distribution on the space of the strategies for player . A *mixed strategy profile* is a collection of mixed strategies for each player.

For example, the dragonfly might decide to move with probability , and stay still with probability ; similarly, the mosquito might decide to move with probability , and stay still with probability .

With mixed strategies, the important value for player to take into account is the *expected utility given the strategic profile*

which we may write when we want to emphasize the role of player .

Now, suppose that the dragonfly decides to set its own paramenter so that its expected utility does not change if the mosquito decides to move or to stay: this corresponds to the dragonfly maximizing its expected utility, given the mixed strategy of the mosquito. Our table tells us that this corresponds to

which has solution . In turn, if the mosquito sets its own parameter so that its own expected utility does not change if the dragonfly decides to move or stay, then

which has solution . The situation where the dragonfly moves with probability and the mosquito moves with probability is a situation none of the two insects has any advantage to change on its own part, given the choice of the other.

**Definition 3.** A *mixed strategy Nash equilibrium* for a game in normal form is a mixed strategy profile such that, for every player and every mixed strategy feasible for player , it is the case that .

And here comes Nash’s great result:

**Nash’s theorem.** Every game in normal form that allows at most finitely many *pure* strategic profiles admits at least one, *possibly mixed* strategy, Nash equilibrium.

It is actually sufficient to prove Nash’s theorem (as he did in his doctoral thesis) when there are only many players, and each of them only has finitely many pure strategies: such limitation is only apparent, because the condition that pure strategy profiles are finitely many means that all players have finitely many pure strategies, and at most finitely many of them have more than one.

The idea of the proof, which we might go through in a future Theory Lunch talk, goes as follows:

- Identify the space of mixed strategic profiles with a compact and convex set for suitable .
- For define a family of continuous transformations .
- By the Brouwer fixed-point theorem, for every there exists a mixed strategic profile such that .
- As is compact, the sequence has a limit point .
- By supposing that is not a mixed strategy Nash equilibrium we reach a contradiction.

We remark that Nash equilibria are not optimal solutions: they are, at most, *lesser evils for everyone given the circumstances*. To better explain this we illustrate a classic problem in decision theory, called the *prisoner’s dilemma*. The police has arrested two people, who are suspects in a bank robbery: however, the only evidence is about carrying firearms without license, which is a minor crime leading to a sentence of one year, compared to the ten years for bank robbery. So, while interrogating each suspect, they propose a bargain: if the person will testify against the other person for bank robbery, the police will drop the charges for carrying firearms without license. The table for the prisoner’s dilemma thus has the following form:

Quiet | Speak | |

Quiet | ||

Speak |

Then the situation where both suspects testify against each other is the only pure strategy Nash equilibrium: however, it is very far from being optimal…

]]>I wrote a post on this on my blog on cellular automata.

Link: http://anotherblogonca.wordpress.com/2014/05/15/random-settings-in-cellular-automata-machines/

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