The quantum harmonic oscillator in the plane

This post investigates some solutions of the (or rather a) quantum harmonic oscillator in the plane. Here the plane is which will be identified with through . Let and denote the partial derivative operators with respect to and respectively. Then the Laplace operator is . The time independent form of the Schrödinger equation for the quantum harmonic oscillator considered here is

where is a complex function and a real eigenvalue (related to energy in physical terms). The operator is called the Hamiltonian and will be denoted by . Only radial solutions will be considered. A complex function is called radial when it is of the form

or

for some real analytic function and some . Radial functions of the first and second form are said to be of degree and respectively. If is a radial function of degree and then .

Working over it is convenient to rewrite the Laplace operator as follows. Let and . If is a complex differentiable function then and while and . (These are the Cauchy-Riemann equations.) The operators and commute and are related to the Laplace operator by

The Hamiltonian is therefore and we will use it in this form. For a radial function

.

Applying the Hamiltonian results in

and in particular if then . So for each degree the function is an eigenfunction of with eigenvalue . Complex conjugation shows that the same holds for of degree . For each degree we found a radial solution for the harmonic oscillator of degree and eigenvalue . These examples do not exhaust all such solutions however. Others can be found by a clever trick that was already known in this context by Schrödinger and Dirac.

This trick is known as the factorisation or algebraic method and the auxiliary operators that appear are called ladder operators or annihilation and creation operators. The following facts can be readily verified:

1. The operators and lower the degree of a radial function by .
2. The operators and raise the degree of a radial function by .
3. If is a radial function of degree then and

   
   
   
   

Combining these observations we find for a radial function of degree :

In particular if is a solution with eigenvalue then is a solution of degree and eigenvalue . In this case and so if . Under the assumption that we can assemble the following table of solutions based on :

The first two operators in this table are called raising or creation operators: they raise the eigenvalue or create energy. The last two are called lowering or annihilation operators for similar reasons. Starting from the solution of degree and eigenvalue and repeatedly applying the operators and we find non-zero solutions for all pairs for which is odd and . This process results in the solutions below up to scalar multiples. Solutions for negative degrees can be obtained by complex conjugation.

Bounds for central binomial coefficients

Let be a positive integer. The binomial coefficient is called a central binomial coefficient. Several bounds for these coefficients are known and the more advanced ones are commonly derived from Stirling’s formula for factorials or from Wallis’ product for . This post presents an alternative and self-contained elementary proof of the bounds

.

Define a function by

and take . From and it follows that

.

So is non-increasing on this interval and therefore or

.

This inequality clearly also holds at the endpoint and since both sides are symmetric in it holds throughout the interval . (It suggests that closely resembles a normal distribution on this interval.) Integration of the inequality above leads to

.

Both the far left and right hand sides of these inequality can be computed explicitly. Starting with the right hand side let

.

Then

And so . To evaluate the left hand side let

By explicit computation we find and . The other values can be found by a recursive relation that follows from partial integration. For positive we have

and therefore the recursion . Using this recursion one can check that the even and odd entries of the sequence are given respectively by

.

Putting all results sofar together we find for positive even integers

and for odd integers

.

Fixed points of entire functions

This story has a long personal history. In my first attempted complex analysis exam I was asked if the entire function has any fixed points. I had no idea how to approach this question and I failed the exam. This first failure was soon overcome but that particular question was —much to my advantage— never addressed during my study again. Much later I found several ways to solve this question.

The image of is invariant under a shift . The “Little Picard” theorem then asserts that this function maps onto since it cannot omit just a single value and so it must have a root. Another way uses “Great Picard” and the fact that only has a single root so it must attain the value infinitely often. Both approaches seem nice enough but depend on non-trivial theorems (“Great Picard” more so than “Little Picard”) and there is no indication of the location of the fixed points. A more pedestrian approach shows that there must be infinitely many fixed points along the curve but this is hardly in the spirit of complex analysis. This was the state of affairs until this week. Then I found a simple and much more satisfying answer.

The Banach fixed point theorem asserts that every contraction of a complete metric space has a single fixed point. I will use that theorem in the following setting.

Let be a non-empty closed convex subset of an open set and let be holomorphic. If there exists a constant such that on then restricted to is a contraction and therefore has a single fixed point in .

For let denote the horizontal strip

and let be the branch of the logarithm that maps the slit complex plane onto the interior of . If then

on so the conditions of the theorem above apply and therefore has a single fixed point (which must lie in the interior.) Each is a fixed point of . Unfortunately the same argument does not work for where has two more fixed points.

The fixed point method works equally well for a number of other functions. Here are two more examples:

Example 1. Let and . The branch of that maps the (closed) upper half plane onto the half strip

is a contraction on . So has exactly one fixed point in each such strip (and therefore also exactly one in its complex conjugate).

Example 2. Let and odd. The branch of that maps the complement of the open unit disc into the strip

is a contraction on . So has exactly one fixed point in each such strip.

These example for , and work so well because their inverses have a nice derivative that is less than one except in a small bounded region. Put in another way these functions all satisfy a differential equation of the form for some and some polynomial .

Acceleration of alternating series

Let be a positive decreasing function such that . Then the alternating series

converges. Depending on the rate at which decays it may however take many terms to get a decent approximation of the sum. Examples of such slowly converging series are

A well known method to accelerate the convergence of such alternating series uses the binomial transform of the function as follows. For let be the -th difference of :

Then it turns out that

and the right hand side may converge much faster. The binomial transform for the two examples above can be computed explicitly to obtain

Both transformed series indeed converge much faster than the original ones. You may recognize the first as the series for taken at . The second one is harder to spot but it turns out to be the series of

taken at .

The binomial transform is a great tool if you can find an explicit expression for it such as in both examples above. This is however not a trivial transform in general. If you do not know an explicit expression for the binomial transform then it is also not very convenient to compute: depends on the values for all .

There is however another much simpler method that only involves the repeated difference for a single fixed value of . This difference is now easy to compute (numerically) since depends only on consecutive values of . It is convenient for the sake of notation to extend to by setting for all . Then for all .

Now the following equalities hold in which the second and third sums are taken over :

If all terms are positive for then partial sums increase for the second series and decrease for the third from onward. This means that these partial sums are lower and upper bounds for the series which is useful to estimate the remaining error for these partial sums.

As an example take . The series for and now become

Game trees under a partial order part 3: Alpha Beta pruning

In this last post of this series of three I will discuss alpha beta pruning. The context will be evaluation functions on game trees that take values in a distributive lattice. Of course a prime example would be the linear lattice over the partially ordered set but this post does not depend on specific features of this linear lattice. The result is a description of alpha beta pruning for distributive lattices that is a slightly improved version of that given by Ginsberg et al in their paper “Alpha Beta pruning under partial orders” from 2001.

Alpha Beta pruning is a way to compute the value of an evaluation function on a game tree for the root node by visiting as little child nodes as necessary. See the previous post for a more extensive introduction to evaluation functions. Here I will only briefly recall the recursive definition of an evaluation function on a game tree that takes values in a lattice . I will assume that has a minimal and maximal element denoted by and . The normal forms and of the linear lattice are examples of this.

1. The value of for a leaf node is given by some heuristic game evaluation function (e.g. chess piece values).
2. At an inner min node (where the minimizing player is to make a move) the value of is defined recursively as the greatest lower bound in of all values of on direct child nodes.
3. At an inner max node the value of is defined as the least upper bound of all values of on direct child nodes.

This definition leads to the following algorithm for the computation of . This algorithm is also the basis of further considerations on ways to prune parts of the game tree that are irrelevant for the computation of on the root node.

function f(node)
  local value
  if node is a leaf node then
    value = heuristic position value of node
  elseif node is a min node then
    value = 1
    for child in all children of node do
      value = value ∧ f(child)
    end
  else
    value = 0
    for child in all children of node do
      value = value ∨ f(child)
    end
  end
  return value
end

The structure of this naive algorithm is simple. To compute the value at a min node start with the maximal value then iterate over all child nodes and adjust the node’s value by taking the greatest lower bound with the value of a child node. At a max node the procedure is similar but starts off with the minimal value and repeatedly takes the least upper bound. The computation recurses depth first left to right over all child nodes, where “left to right” is according to the enumeration order of child nodes. Move ordering does not play a role in this naive algorithm but it can play a big role in alpha beta pruning in terms of the number of pruned nodes. I will not address move ordering in this post. During execution of the algorithm the state of the computation is schematically depicted below. As in the example in the previous post I will assume that the top node is a min node.

Intermediate values in the depth first left to right computation of an evaluation function

Here and are the intermediate values that are computed for the inner min and max nodes respectively. The symbol indicates all subtrees located to the left of the current path of computation in the tree. These are the subtrees for which a value is already (recursively) computed. When the computation progresses each intermediate value descends from and each ascends from . When we would cut the computation short at the point depicted above then the algorithm would track back up and yield the final outcome

at the root node. The essential point of alpha beta pruning is that it identifies when this intermediate value remains constant even when gets larger in the remainder of the computation. Because in that case any further children of the node marked in the diagram will not contribute to the final result and can be pruned (skipped). In the picture above the search path ends with a max node but that is irrelevant for alpha beta pruning as we will see.

A lattice has a canonical partial order where is equivalent to (or alternatively ). The lattice operators and indeed result in the greatest lower bound and least upper bound with respect to this ordering. The following lattice property is the key point in alpha beta pruning.

Lemma: When and then and .

Since the greatest lower bound is also a lower bound for and and is therefore less than or equal to the greatest lower bound . The other inequality is derived similarly. For sequences and in define sequences and recursively by

1. and is obtained from by the substitution .
2. and is obtained from by the substitution .

So the sequence starts with

and the sequence with

The following inequalities are a direct consequence of our lemma above. The sequences and satisfy

for all indices . In particular is an ascending sequence, is a descending sequence, each is a lower bound for and each is an upper bound for . If or for some index then both sequences are constant from that point onwards. The values in the sequences and are precisely those obtained from aborting the computation of the evaluation function for the root node at min node (sequence ) or at max node (sequence ). Hence further nodes can be pruned from the computation as soon as both sequences yield equal values.

Alpha Beta pruning (strong version): At the first index where either or further subnodes of max node or min node can be pruned.

Note that this formulation of alpha beta pruning resembles “classical” alpha beta pruning for totally ordered sets in that we have an ascending sequences of high values and a descending sequence of low values and nodes can be pruned as soon as these values coincide. There is also an unfortunate difference between the two. In classical alpha beta pruning the high and low values are easy to compute progressively as a simple maximum or minimum value. In contrast high and low values in the pruning algorithm above are not so simple to compute. Since they are computed deepest node first it requires constant back tracking. Moreover, computation of and can be much more expensive than taking simply the maximum or minimum of two numbers.

Also note that the distributive property of the lattice is not used at all in this version of alpha beta pruning! This does not contradict the results of Ginsberg et al since they formulate alpha beta pruning (in particular “deep pruning”) differently. I will briefly come back to this point at the end of this post.

There is a weaker version of alpha beta pruning based on alternative sequences and that are easier to compute. The consequence is however that it may take longer before the ascending high values and descending low values allow cut off, resulting in less aggressive pruning. The weaker estimates for high and low values follow from the following identities (where the distributive property of is now essential). Let and define two new sequences and by

and
.

So is the least upper bound of the values of all max nodes above min node and is the greatest lower bound of the values of all min nodes above max node . If we take and then for all indices the following equalities hold:

and
.

These can be proved by induction and the distributive property of the lattice . Taking in the first equation and in the second results in

and
.

Comparing these equalities it follows that implies and implies . This leads to the following version of alpha beta pruning.

Alpha Beta pruning: If the value of min node gets less than or equal to or the value of max node gets greater than or equal to then further child nodes can be pruned.

This is a slightly different formulation of alpha beta pruning than Ginsberg et al use in their paper. What they call (shallow or deep) pruning comes down to either for some at min node or for some at max node . These are stronger conditions than those we found above and will therefore prune fewer nodes in general.

Here is an algorithmic formulation of alpha beta pruning. The value of the root node is computed by f(root, 0, 1).

function f(node, α, β)
  local value
  if node is a leaf node then
    value = heuristic position value of node
  elseif node is a min node then
    value = 1
    for child in all children of node do
      value = value ∧ f(child, α, β ∧ value)
      if value ≤ α then break end
    end
  else
    value = 0
    for child in all children of node do
      value = value ∨ f(child, α ∨ value, β)
      if value ≥ β then break end
    end
  end
  return value
end

To sum up the results of this post: I described two versions of alpha beta pruning that allow to prune nodes from the computation of an evaluation function with values in a distributive lattice at the root node. The first, strong, version has the disadvantage that it is computationally more difficult than the second. Both versions are stronger (potentially prune more nodes) than the method described in the paper by Ginsberg et al.

Game trees under a partial order part 2: Evaluation functions

The first part of this series introduced the maximizing and minimizing orders on finite antichains in any partially ordered set . In this second part I will show how these orders and their properties are used to define evaluation functions on game trees. The previous post was mainly based on the paper “Searching game trees under a partial order” by P. Dasgupta et al. In this second post I will show how it is related to another paper “Alpha-Beta pruning under partial orders” by M. Ginsberg et al from 2001. This second paper has an interesting different approach. Instead of starting with a partial order from the beginning it investigates what structure an evaluation function on a game tree must have to satisfy some reasonable constraints. The answer they find is that such an evaluation function must be defined on a lattice, that is, a set with operators “greatest lower bound” and “least upper bound” . An evaluation function is then defined recursively, starting at the leaves. The minimizing player combines the values of subnodes with while the maximizing player uses . Moreover, Ginsberg et al show that such an evaluation function allows alpha beta pruning (which I will define in detail in the next post) exactly if the lattice is distributive. While the paper by Ginsberg references that of Dasgupta the relation between the two is not made very explicit. This post will give a very precise and (I think) satisfying relation between the two papers ultimately improving the results of both: The work of Dasgupta is formalized in terms of lattice theory and can now be applied to any partially ordered set, not just and the result of Ginsberg on alpha beta pruning is sharpened in the next post, potentially leading to more aggressive pruning in game tree searches.

Consider a game tree in which a min and max player make moves in turn. Suppose that leaf nodes in this tree get a value in the partially ordered space assigned to them. Is there a way, given the values at the leaves, to define an evaluation function on the entire tree? Let’s first assume that both players in fact use an (unspecified) linear extension of the partial order to select their optimal move. (Dasgupta et al call such a linear extension a strategy. See the intermezzo below for why it is fair to assume that both players use the same strategy.) Here is an example how this linear extension can be used to define an evaluation function on a tree of positions. The symbols “” at the leaf nodes indicate some (not necessarily equal) values in . The greatest lower bound and least upper bound and are taken with respect to the linear extension .

An evaluation function based on a linear extension (or strategy)

In this example the minimizing player is to move and the tree represents possible move developments that are considered. The value at inner nodes is defined recursively based on the linear extension: The minimizing player takes the minimum (the greatest lower bound) of the subnodes while the maximizing player takes the maximum (the least upper bound). This leads to a value for the top node in and the minimizing player will select a move that leads to this minimum.

Intermezzo. Assuming that the min player in this example uses an unspecified linear extension (or “strategy” as Dasgupta calls them) to select moves is central in our considerations. But why is it fair to assume that the max player uses the same strategy? The reason is simple. Suppose the max player uses a different strategy or in fact any system at all (even random decisions) to select moves. In that case the altered evaluation function would assign to each node a value that is less than or equal to the value obtained from equal strategies (when compared with the min player’s linear extension). This follows from the recursive definition of the evaluation function. Since the min player’s goal is to reach a minimal value, the worst case scenario is to assume that the opponent uses the same strategy. End of intermezzo.

Any evaluation function defined as above still depends on the choice of linear extension, so different linear extensions give different evaluation functions. What we are looking for instead is a way to define a single evaluation function based only on the given partial order on . There is a nice trick to turn the example above into exactly such a function, but it requires to alter the target space. To define the new target space start with , the space of all functions that map linear extensions of to . The evaluation defined above at any node can be regarded as an example of such a function by treating the linear extension in its definition not as fixed but as a variable. Let me make this more precise based on the following observations:

1. There is a canonical embedding as constant functions: for all linear extensions .
2. If then define an element by where in the right hand side refers to the minimum with respect to .
3. If then define an element by where in the right hand side refers to the maximum with respect to .
4. Since any totally ordered set is a distributive lattice with respect to its minimum and maximum operators and it follows that the definitions above turn into a distributive lattice and therefore also into a partially ordered set.

The entire space is too big for our purpose, therefore I define , the linear lattice over as the smallest sublattice of that contains . Then is still a distributive lattice. The linear lattice will serve as the target space for evaluation functions on game trees under the partial order .

Remark. The set embeds into as a partially ordered set but not always as a lattice. For example if have a greatest lower bound then is a lower bound for and but not necessarily the greatest lower bound. End of remark.

Example. If is a totally ordered space then . End of example.

An evaluation function on a game tree is now defined recursively, as in the picture above, as follows.

1. Assign values in to the leaf nodes based on some heuristics.
2. Define the value of an inner min node as the greatest lower bound in of the values of all its direct successors.
3. Define the value of an inner max node as the least upper bound in of the values of all its direct successors.

Note that this construction is completely in line with the results of Ginsberg et al. The fact that is a distributive lattice even implies that alpha beta pruning should be applicable in the recursive computation of the evaluation function. I will address that part in the next post.

The linear lattice may seem rather abstract but the fact that it is distributive and generated by means that each element can be put in both a disjunctive and conjunctive normal form built from elements of only. These normal forms can be described directly in terms of minimizing and maximizing orders as introduced in the first post. This will identify the linear lattice with what Dasgupta et al call “dominance algebra”. Define two partially ordered spaces, suggestively called and , by applying the star order extension twice as in the picture below:

Two step construction of DNF (top) and CNF (bottom)

So is the set of finite antichains in ordered by their maximizing order and the set of finite antichains in ordered by their minimizing order. If is a linear extension of I denote it’s minimum and maximum operators by and respectively to distinguish them from the lattice operators in CNF and DNF. The following statements identify the normal forms of .

CNF: For any partially ordered space the space is a lattice with minimal and maximal elements and respectively. The sublattice is isomorphic to via the rule .

DNF: For any partially ordered space the space is a lattice with minimal and maximal elements and respectively. The sublattice is isomorphic to via the rule .

I will only show the CNF case. In the previous post we saw that any maximizing order always has a greatest lower bound operator and it also has a least upper bound operator if its base space has one. Since in particular always has a least upper bound operator, this shows that is a lattice. The empty set (which is a finite antichain) is maximal for any maximizing order and minimal for any minimizing order. Therefore the elements and are minimal and maximal elements for this lattice respectively. A simple verification shows that the given evaluation rule at least defines a surjective homomorphism onto so it remains to prove that this homomorphism is also injective (meaning that the conjunctive normal form is unique). Suppose are such that for all linear extensions . I will show that this implies (which suffices since by symmetry then also and ).

Let . For any there exists a linear extension such that

1. is maximal in the lower set and
2. if and then .

This implies that since, being an antichain, no element of other than itself can be entirely contained in and therefore must have a maximum greater than . Then by assumption also . In particular all with must contain and at least one such element must exist in . Since this holds for any it follows that for all with . This implies that and hence .

Summing up the results of this post: The linear lattice is the natural target space for evaluation functions on game trees based on heuristic evaluation of positions with values in the partially ordered set . Both lattices and (minus their maximal and minimal elements) are isomorphic to the linear lattice and explicitly represent elements in their unique conjunctive and disjunctive normal form respectively. In particular the CNF and DNF lattices are themselves distributive and isomorphic. The isomorphisms both ways correspond to what in the paper by Dasgupta et al is called “MAX to MIN” and “MIN to MAX”.

In the next and final post in this series I will discuss alpha beta pruning for game trees under a partial order.

Game trees under a partial order part 1: Partial orders

I’ve been interested in the subject of alpha-beta pruning of game trees for quite some time now. It has always struck me as unsatisfactory however that it relies on a total order of game positions (in the form of an evaluation function). It would be much more natural to evaluate game positions based on several distinct features and select best moves on this richer set of information. This means essentially that the total order of game positions is replaced by a partial order. There are a number of articles that explore this idea. The theory is, somewhat surprisingly maybe, quite a bit more involved than it is for total orders. The problem is that with a partial order there is now often not a single best move to choose from and the bookkeeping to handle all these choices is rather intricate. This first post in a series of three is based on a foundational paper “Searching game trees under a partial order” by P. Dasgupta et al, published in 1996. The motivation for these posts was firstly because it is a fascinating subject but also because I found the paper rather difficult to read and understand.  I think that a more mathematical style would help the overall presentation of the subject and  that is what I will try to accomplish in this series of blogs. This first post concentrates on the concept of partial orders underpinning the concept of game tree evaluations later on. That said, this post is also interesting in itself as a result in the theory of partial orders, even without the context of its application in game trees.

Let be a partially ordered set. It suffices to consider only finite sets for our application but the presented theory applies to any set and we will not assume finiteness. I will write as shorthand for and . Our goal in this post is to introduce a sensible way to compare finite (possibly empty) antichains in . An antichain is a subset of in which any two distinct elements are unordered. Recall that a linear extension  of is a total order on that is consistent with in the sense that   implies . Throughout this series of posts the symbol always indicates a linear extension and and denote the minimum (greatest lower bound) and maximum (least upper bound) for this linear extension respectively.

A subset is called a lower set if for all and the inequality implies that . Any subset can be extended to a smallest lower set containing by the definition

for some .

Note that if are antichains such that then . Similarly one can define upper sets and the smallest upper set containing .

Lemma: Let be nested lower sets. Then there is a linear extension of such that for all indices and all elements and the strict inequality holds.

If we take then this lemma simply states that there exists a linear extension. A more important consequence is the following.

Corollary: If is an antichain then there exists a linear extension such that for any and the inequality implies that . Moreover, for any this extension can be chosen such that is maximal in .

The first part follows by taking and the lower set in the lemma. For the second part take and the two lower sets and . Since it is greater than all other elements in for the resulting linear extension.

Linear extensions are in a sense the “hidden variables” in the theory of game trees under a partial order. If game positions are only partially ordered then searching for moves will most likely not result in a single best move but in a finite set of best moves. Nevertheless a player must still choose a single move from such a set. I will make the fundamental assumption that both players use an otherwise unspecified linear extension to make their choice. (Dasgupta et al call the linear extension they use a “strategy”. Moreover they consider situations where both players use a different strategy. This turns out to be unnecessary as we will see in the next post.) Then the “minimizing player” MIN always selects a minimum move according to while the “maximizing player” MAX always selects a maximum move. Clearly different linear extensions lead to different gameplay for both players but nevertheless this idea will lead to a definition of gameplay that only depends on the underlying partial order and not on a particular choice of strategy. In what follows the central idea is therefore to single out properties that hold for all possible linear extensions and are “universal” in that sense.

Based on this idea we will construct minimizing and maximizing orders on finite antichains of . Suppose that player MIN is in a situation where only two moves are possible. The first move will lead to a finite set of best moves for player MAX to choose from while the second move leads to a finite set . Then player MAX will select the maximum when MIN plays the first move and when MIN plays the second. For player MIN this means that the first move is preferred over the second precisely when . If this relation holds for all possible linear extensions then the first move for MIN is universally better than the second. We can take this as the definition for a minimizing order on finite antichains in and express it as . This definition is potentially rather awkward, quantifying over all linear extensions, but it can be drastically simplified as I will now show.

If then for each there is a such that . In particular, for any linear extension the inequality also holds and therefore . So holds in this case. Now assume instead that there is an such that . Then according to our lemma there is a linear extension such that for all and so for this particular linear extension. In this case the relation does not hold. This justifies the definition below. An alternative formulation that works for all finite subsets is included as a small bonus. See the intermezzo for some more background.

Minimizing order: For finite antichains define the relation as or alternatively for all finite subsets as .

Similar considerations (with the roles of the MIN and MAX player reversed) lead to the following definition of a maximizing order, where now indicates the smallest upper set containing :

Maximizing order: For finite antichains define the relation as or alternatively for all finite subsets as .

The minimizing order expresses that the minimizing player MIN prefers the set of game positions over while the maximizing order expresses that player MAX prefers over . It is not difficult to check that the maximizing and minimizing orders are in fact partial orders on finite antichains in . Note that when are finite antichains and then and . In particular for any finite antichain the inequalities and hold. This also shows that these are different partial orders in general.

Intermezzo. The alternative formulation of the maximizing and minimizing orders results in partial orders on all finite subsets. These partial orders are then almost but not quite identical to what Dasgupta et al call “procedure compare.” Where Dasgupta et al prefer a non-empty set over an empty one in that procedure, the maximizing and minimizing orders always prefer the empty set over a non-empty one as already stated before. The transitivity property requires some attention and can be proved by the nice picture below, where it is assumed that and .

Descending chains in the transitivity proof

The nodes in the diagram correspond to subsets of where the letters in a node select all elements that are exactly in these subsets and in no others. So “A” means and “A, B” means and so on. The top two nodes together form the set and the bottom two nodes . Directed edges leaving a node indicate that for each in the source set there exists in one of the target sets such that . The edges together encode the information obtained from the relations and . Since descending chains must be finite each chain starting in (top row) must eventually reach (bottom row) and transitivity follows. End of intermezzo.

Until now I only considered the conditions under which a single antichain is preferred over an antichain for player MIN, which is expressed by . However, player MIN will often have to choose from more than two possible moves so we also have to consider conditions under which a set of more than one antichain is preferred over . That would mean that for any linear extension there is some index such that where the index might depend on the choice of linear extension. However, it turns out that in this case there must be an index such that already . In other words, player MIN prefers a set of antichains over if and only if some member of that set is already preferred over . This again follows from the lemma. Let be a linear extension such that for all and . Suppose that for all indices . Then for each index there is an such that . Then for all indices and therefore the set is not preferred over .

Let denote the set of finite (possibly empty) antichains in the partially ordered set .  Then both and are partially ordered sets with the maximizing and minimizing order respectively. Later on the lattice structure of such partial orders will play an important role. A partially ordered set is a lattice when all non-empty finite subsets have a greatest lower bound and a least upper bound . In that case it is easy to see that and are uniquely defined. The following theorems give sufficient conditions for and to be lattices.

Maximizing lattice: In the partially ordered set every non-empty finite subset has a greatest lower bound . If every non-empty finite subset has a least upper bound then the same holds for and so is a lattice.

Minimizing lattice: In the partially ordered set every non-empty finite subset has a least upper bound . If every non-empty finite subset has a greatest lower bound then the same holds for and so  is a lattice.

I will prove only the maximizing lattice property, so the partial order on is the maximizing order in what follows. Let be finite antichains. The union is a finite subset of but not necessarily an antichain.  Let be the set of minimal elements in , that is, those elements which have no smaller elements in . Then is an antichain and because is finite for all indices . So is a lower bound for . If is any lower bound then for each there is a such that . This means that since and so is the greatest lower bound.

Now suppose every non-empty finite subset has a least upper bound . If is an upper bound for then for all there exist such that for all indices and so . Let be the set of minimal elements

.

Then is an antichain and by the above. Now is itself also an upper bound for and so it is the least upper bound. This shows that under the given condition is a lattice.

To sum up the results of this post: For any partially ordered set we constructed a minimizing and a maximizing partial order on the set of finite antichains in . These indicate when a set of outcomes is universally preferred (under all linear extensions) over another set by a minimizing and maximizing player respectively. For these “star extended” partially ordered sets we defined the lattice operations of taking the greatest lower bound and least upper bound (when these exist). In the next post in this series we will start to explore actual game trees under a partial order, starting with the definition of evaluation functions.