Every hierarchy of beliefs is a type

When modeling game situations of incomplete information one usually considers the players' hierarchies of beliefs, a source of all sorts of complications. Hars\'anyi (1967-68)'s idea henceforth referred to as the "Hars\'anyi program" is that hierarchies of beliefs can be replaced by "types". The types constitute the "type space". In the purely measurable framework Heifetz and Samet (1998) formalize the concept of type spaces and prove the existence and the uniqueness of a universal type space. Meier (2001) shows that the purely measurable universal type space is complete, i.e., it is a consistent object. With the aim of adding the finishing touch to these results, we will prove in this paper that in the purely measurable framework every hierarchy of beliefs can be represented by a unique element of the complete universal type space.Comment: 19 page

Generalized Descents and Normality

We use Janson's dependency criterion to prove that the distribution of $d$-descents of permutations of length $n$ converge to a normal distribution as $n$ goes to infinity. We show that this remains true even if $d$ is allowed to grow with $n$, up to a certain degree.Comment: 7 page

Lower Bounds for RAMs and Quantifier Elimination

We are considering RAMs $N_{n}$, with wordlength $n=2^{d}$, whose arithmetic instructions are the arithmetic operations multiplication and addition modulo $2^{n}$, the unary function $\min\lbrace 2^{x}, 2^{n}-1\rbrace$, the binary functions $\lfloor x/y\rfloor$ (with $\lfloor x/0 \rfloor =0$), $\max(x,y)$, $\min(x,y)$, and the boolean vector operations $\wedge,\vee,\neg$ defined on $0,1$ sequences of length $n$. It also has the other RAM instructions. The size of the memory is restricted only by the address space, that is, it is $2^{n}$ words. The RAMs has a finite instruction set, each instruction is encoded by a fixed natural number independently of $n$. Therefore a program $P$ can run on each machine $N_{n}$, if $n=2^{d}$ is sufficiently large. We show that there exists an $\epsilon>0$ and a program $P$, such that it satisfies the following two conditions. (i) For all sufficiently large $n=2^{d}$, if $P$ running on $N_{n}$ gets an input consisting of two words $a$ and $b$, then, in constant time, it gives a $0,1$ output $P_{n}(a,b)$. (ii) Suppose that $Q$ is a program such that for each sufficiently large $n=2^{d}$, if $Q$, running on $N_{n}$, gets a word $a$ of length $n$ as an input, then it decides whether there exists a word $b$ of length $n$ such that $P_{n}(a,b)=0$. Then, for infinitely many positive integers $d$, there exists a word $a$ of length $n=2^{d}$, such that the running time of $Q$ on $N_{n}$ at input $a$ is at least $\epsilon (\log d)^{\frac{1}{2}} (\log \log d)^{-1}$