Finite promise games

Friedman's finite promise games is a family of two-player games defined by Harvey Friedman.

As the name implies, each of games is finite, and its length is actually predetermined. At each turn, one of the players (which we will call Alice) chooses an integer or a number of integers (an offering) and the other player (which we will call Bob) has to make promise restricting his future possible moves. Bob wins if and only if he can keep all the promises until the end of the game.

Finite piecewise linear copy/invert games
Call a function \(T: \mathbb{N}^k \mapsto \mathbb{N}\) piecewise linear iff it can be expressed using finitely many affine functions, using inequalities with integer coefficients to separate the function's pieces. Given a piecewise linear function \(T\), we say that a vector \(\mathbf{y} \in \mathbb{N}^k\) is a \(T\)-inversion of \(x\) iff \(T(\mathbf{y}) = x\).

Given a piecewise linear function \(T: \mathbb{N}^k \mapsto \mathbb{N}\) and two positive integers \(n\) and \(s\), we define the finite piecewise linear copy/invert game \(G(T, n, s)\) as follows. \(G(T, n, s)\) is a two-player game alternating between Alice and Bob, with \(n\) rounds in total. Alice goes first.

On her turn, Alice selects a number \(x \in [0, s]\), either of the form \(w!\) or \(y + z\), where \(y\) and \(z\) were previously played by Bob. \(x\) is called her offer. Bob either has the choice of accepting or rejecting the offer:


 * By accepting, he plays \(x\), and promises that none of the integers he ever plays will contain a \(T\) inversion of \(x\).
 * By rejecting, he plays a \(T\) inversion of \(x\) of his choice, and promises that none of the integers that he ever plays are \(x\).

If Bob ever breaks one of his promises, he loses the game. If Bob lasts the entire \(n\) rounds without ever breaking a promise, he wins. (Promises apply to all past, present, and future plays within the game.)

Bob always has a winning strategy for any given FPLCI game.

In fact, he can still win even if we restrict the offers that he is allowed to reject. Consider the modified game \(G_m(T, n, s)\) where Bob is forced to accept all factorial numbers greater than \(m\). Bob can win \(G_m(T, n, s)\) for all \(T\) and \(n\) and sufficiently large \(m\) and \(s\).

Let \(FLPCI(a)\) be the minimal \(N\) such that Bob can win \(G_m(T, n, s)\) for all \(T\), all \(n < a\), and all \(m, s > N\).

Finite polynomial copy/invert games
All polynomials discussed in this section have the signature \(\mathbb{Z}^k \mapsto \mathbb{Z}\) (for a fixed \(k\)) and have only integer coefficients.

Given a polynomial \(P\), we say that a vector \(\mathbf{y}\) is a special \(P\)-inversion of \(x\) iff \(P(\mathbf{y}) = x\) and every component of \(\mathbf{y}\) is strictly between 0 and \(x/2\).

Given two polynomials \(P\) and \(Q\) and two positive integers \(n\) and \(s\), we define the finite polynomial copy/invert game \(G(P, Q, n, s)\) as follows. \(G(P, Q, n, s)\) is a two-player game alternating between Alice and Bob, with \(n\) rounds in total. Alice goes first.

On her turn, Alice selects a number \(x \in [-s, s]\), of the form \((w!)!\) or \(P(\mathbb{y})\) or \(Q(\mathbf{y})\), where \(\mathbf{y}\) is a vector in \(\mathbb{Z}^k\) whose components were all integers previously played by Bob. \(x\) is called her offer. Bob either has the choice of accepting or rejecting the offer:


 * By accepting, he plays \(x\), and promises that none of the integers he ever plays will contain a special \(P\) or \(Q\) inversion of \(x\).
 * By rejecting, he plays a special \(P\) or \(Q\) inversion of \(x\) of his choice, and promises that none of the integers that he ever plays are \(x\).

If Bob ever breaks one of his promises, he loses the game. If Bob lasts the entire \(n\) rounds without ever breaking a promise, he wins. (Promises apply to all past, present, and future plays within the game.)

Bob always has a winning strategy for any given FPCI game.

In fact, he can still win even if we restrict the offers that he is allowed to reject. Consider the modified game \(G_m(P, Q, n, s)\) where Bob is forced to accept all factorial numbers greater than \(m\). Bob can win \(G_m(P, Q, n, s)\) for all \(P,Q,n\) and sufficiently large \(m\) and \(s\).

Let \(FPCI(a)\) be the minimal \(N\) such that Bob can win \(G_m(P, Q, n, s)\) for all \(P\) and \(Q\), all \(n < a\), and all \(m, s > N\).

Finite order invariant copy/invert games
Let \(R : \mathbb{N}^{2k} \mapsto \{0, 1\}\) be order invariant &mdash; that is, the order of its inputs does not affect its output. Given a vector \(\mathbf{x} \in \mathbb{N}^{k}\), we say that a vector \(\mathbf{y} \in \mathbb{N}^{k}\) is an R-inversion of \(\mathbf{x}\) iff \(\max(\mathbf{y}) < \max(\mathbf{x})\) \(R(\mathbf{x}, \mathbf{y}) = 1\) (where the components of the two vectors are concatenated into a vector with \(2k\) components, which is the argument to R).

Given order-invariant \(R : \mathbb{N}^{2k} \mapsto \{0, 1\}\) and two positive integers \(n\) and \(s\), we define the finite order invariant copy/invert game \(G(R, n, s)\) as follows. \(G(R, n, s)\) is a two-player game alternating between Alice and Bob, with \(n\) rounds in total. Alice goes first.

On her turn, Alice selects an integer vector \(\mathbf{x} \in [-s, s]^k\), each component being of the form \(z\) or \(z + 1\) or \(w!\), where \(z\) is an integer previously played by Bob. \(\mathbf{x}\) is called her offer. Bob either has the choice of accepting or rejecting the offer:


 * By accepting, he plays \(\mathbf{x}\), and promises that he never plays an \(R\) inversion of \(\mathbf{x}\).
 * By rejecting, he plays an \(R\) inversion of \(\mathbf{x}\) of his choice, and promises that he will never play \(\mathbf{x}\).

If Bob ever breaks one of his promises, he loses the game. If Bob lasts the entire \(n\) rounds without ever breaking a promise, he wins. (Promises apply to all past, present, and future plays within the game.)

Bob always has a winning strategy for any given FOICI game.

Finite linear copy/invert games
Given two vectors \(\mathbf{x},\mathbf{y} \in \mathbb{N}^k\), we say that they are additively equivalent iff, for all \(i, j \leq k\), \(\sum^i \mathbf{x} < \sum^j \mathbf{x} \Rightarrow \sum^i \mathbf{y} < \sum^j \mathbf{y}\) (where \(\sum^n \mathbf{v}\) is the sum of the first \(n\) components in \(\mathbf{v}\)).

Let \(V\) be a \(p\)-tuple of vectors in \(\mathbb{N}^k\) and two positive integers \(n\) and \(s\), and define the finite linear copy/invert game \(G(V, n, s)\) as follows. \(G(\mathbb{v}, n, s)\) is a two-player game alternating between Alice and Bob, with \(n\) rounds in total. Alice goes first.

On her turn, Alice selects a number \(x \in [0, s]\), either of the form \(w!\) or \(y + z\), where \(y\) and \(z\) are integers previously played by Bob. \(x\) is called her offer. Bob either has the choice of accepting or rejecting the offer:


 * By accepting, he plays \(x\), and promises that \(x\) can never be written as \(\sum \mathbf{y}\), where all the components of \(\mathbf{y}\) were played by Bob at any time and \(\mathbf{y}\) is additively equivalent to a vector in \(V\).
 * By rejecting, he plays \(\mathbf{y}\), where \(\sum \mathbf{y} = x\) and \(\mathbf{y}\) is additively equivalent to a vector in \(V\), and promises that he will never play \(x\).

If Bob ever breaks one of his promises, he loses the game. If Bob lasts the entire \(n\) rounds without ever breaking a promise, he wins. (Promises apply to all past, present, and future plays within the game.)

Bob always has a winning strategy for any given FLCI game.

Functions
As shown by Friedman, the functions \(FPLCI\) and \(FPCI\) are extremely fast growing, exceeding every function provably recursive in theory SMAH, that is, ZFC augmented with, for every \(k\), a sentence "there exists a strongly \(k\)-Mahlo cardinal". It's worth noting that if we replace this infinite set of sentences with a single "for every \(k\) there exists a strongly \(k\)-Mahlo cardinal" we actually get quite stronger theory, denoted SMAH+, in which the functions are provably recursive.