An Example. The basicformalism we use here are logicalimplicationslike p.q => r.tmeaning “when p and q hold, then also r and t hold”. This is often called (definite) Hornclauses.16 We use here a notion from AIFω [22], a noveladd-on for OFMC for denoting the implications, while AIFω rules actually mean statetransitions. For the formulas we are using in this section, however, it does not make a difference. The full language of AIFω will be explained in a specialchapter later.Let us begin with the intruder, and let iknows(m) stand for “the intruder knows m”. We could specify first his initialknowledge like this:=> iknows( a); => iknows( pk( a));=> iknows( b); => iknows( pk( b ));=> iknows( i); => iknows( pk( i)); => iknows( inv( pk( i)));to mean that the intruder knows the agents a, b, i their publickeys and his own private key. Here we use implications without a left-hand side, i.e., the right-hand side factsholdsunconditionally. If we wantto consider more agents, then we would have to make longenumerations. ThereforeAIFω allows to first define some data types:Honest = {a, b}; Dishonest = { i};User = Honest ++ Dishonest;We can then use these types in rules and later change the number of agents without changing any rules and without making long enumerations. To that end, every rule has a rule head of the form rulename(V ariable : Type, V ariable : Type, ...) The type can be any of the user-defined types (like Honest here) or Untyped. Untyped variables, however, have an importantrestriction: every untyped variable must occur in the left-hand side (and may occur in the right-hand side also). In other words, it is not allow to have untyped variables that only occur in the right-hand side.The listing above is then written simply as follows:users( A: User) => iknows( A); publickeys( A: User) => iknows( pk( A ));privatkeys( D: Dishonest) => iknows( inv( pk( D )));The first rule has the name “users” and says that for any A of type User, the intruder knows A. The other rules are similar.Let now• crypt(k, m) stand for {m}k, i.e., the asymetric encryption with key k of message m,• and pair(m1, m2) stand for the pair of m1 and m2.Then the Dolev-Yaomodel for asymmetric encryption and pair is described by the following formulas:asymenc( M1 : untyped , M2 : untyped ) iknows( crypt( M1 , M2 )). iknows( inv( M1 )) => iknows( M2 ); asymdec( M1 : untyped , M2 : untyped ) iknows( M1 ). iknows( M2 ) => iknows( crypt( M1 , M2 )); 16Actually, by definition, Horn clauses can on...

An Example. Consider a one-asset one-period economy, with a zero riskless interest rate. There are three types ofagents: an informed trader (Primus), a single market maker (Secunda), and nine uninformed traders. Among the uninformed traders, some are willing to placeorders (for buying or selling) of 5 units, while the others always place orders only for 1 unit. We call these two types of uninformed traders respectively “big” and “little”.The only available asset is a risky stock which will pay a risky amount Y at the end of the period. We assume that Y is randomly distributed over the interval [0, 4] with unconditionalmean m = 2. The informed trader knows the realization of Y (because he has received a perfectly informativesignal about it), while Secunda does not.Finally, we assume that both the uninformed traders are price sensitive. Any uninformed trader placing an order of 1 is willing to pay up to a price of 2, equal to the unconditional mean. On the other hand, the four big uninformed traders are willing to accept a higherask price if they decide to place a bigorder. In particular, U1 is willingto buy his batch of 5 units up to a price of p = 2.3 each; U2 up is willing to buy it up to a price of p = 2.45 each; U3 up to a price of p = 2.6 each; and U4 up to a price of p = 2.9 each.For simplicity, suppose that Primus knows that the true value ofthe asset is 4 and therefore wants to buy the asset. Consider Secunda’s decision about the ask price. She must a set a price for little orders of 1 unit and a (possiblydifferent) price for big orders of 5.— —We prove that, if Secunda acts competitively, there exists no ask price which can keep the market open. First, assume that Primus trades big. If Secunda acts competitively, she must sets a price for a big trade of 5 such that her expected losses to Primus equal the expected gains from the biguninformed traders. Since there is a one in five chance that the bigtrade comes from Primus, the price p must solve the equation (1/5)(p 4)+ (4/5)(p 2) = 0, from which p = 2.4.However, if Secunda were to quote a price p = 2.4, U1 would not pass his order because the price is too high for him. This would make the number of uninformed big traders — —drop to three, in which case the competitive equilibrium price should solve the equation (1/4)(p 4) + (3/4)(p 2) = 0, from which p = 2.5. At such a price, U2 would drop out rising the new competitive price (with only two big uninformed traders left) to p = 2.67.At such a price, U3 wo...