Equilibrium dominant strategy

The Prisoner s Dilemma equilibrium differs from all the other equilibria mentioned so far, in that it is made up of aaions each of which is the best response to anything others could do, not just to their equilibrium behavior. Equilibrium then does not require a person to have correct expectations about what others will do, since he will take the same action whatever he expects them to do. (An action of this type is called a dominant strategy. 1 If he expects wrongly, he will be surprised, perhaps unpleasantly so, but he will not regret what he did. In such cases there can be unintended consequences in equilibrium. Usually, however, surprise and regret go together. 

Strengthened in a dominant strategy equilibrium, in which truth-revelation is the best-response for an agent whatever the strategies and preferences of other agents. A dominant strategy and IC mechanism is simply called a strategyproof mechanism. Formally ... 

Table 5.2. Double auction mechanisms. The traded column indicates the number of trades executed where I is the efficient number of trades. The equil column indicates whether the mechanism implements a dominant strategy or Bayesian-Nash equilibrium (BNE).

In words, the IC constraint says that assuming other agents truthfully report their types, ones best response is to truthfully report ones type. Phrased differently, truth telling is a (Bayesian) equilibrium of the mechanism. It is sometimes common to require a stronger condition, dominant strategy incentive compatibility, which we discuss later in section . 

Strategy We start with the equilibrium that produces the ion of interest (such as ) and write its activity in terms of the acidity constant for its formation (fCa2). That expression will contain the activity of the conjugate acid (HA ), which we can express in terms of the activity of its conjugate acid (HjA) by using the appropriate acidity constant (K i). This equilibrium dominates all the rest provided the molecule is small and there are marked differences between its acidity constants, so it may be possible to make an approximation at this stage. 

The problem describes a weak diprotic acid and asks for ion concentrations. To determine concentrations of all ions, we need to consider more than one equilibrium. This is done in stages, starting with the dominant equilibrium. We apply the seven-step strategy. The problem asks us for the concentrations of the ions in carbonated water, in which the major species are H2 CO3 and H2 O. 

Strategy Acetic acid is a weak acid consequently, we expect the molarity of H30+ ions to be less than 0.10 moI-L-1 and, therefore, its pH to be greater than 1.0. To find the actual value, we set up an equilibrium table S with the initial molarity of acid equal to 0.10 mol-L 1 and allow the molarity of acid to decrease by x mol-L1 to reach equilibrium. Assume that the presence of acid dominates the pH and therefore that the autoprotolysis of 5 water need not be considered. We assume x is less than about 5% of the ini-j rial molarity of acid and simplify the expression for the equilibrium constant f by ignoring x relative to the initial molarity of the acid. This assumption i must be verified at the end of the calculation. 

STRATEGY Expect pH > 7, because amines are bases. Assume that the presence of base dominates the pH and therefore that the autoprotolysis of water need not be considered. Calculate the molarity of OH ions by using the equilibrium table as explained in Toolbox 10.1, but use Kh instead of fCa. Calculate [OH-], convert it to pOH, and then convert that pOH to pH by using the relation pH + pOH = 14.00. 

In other words, the simplification of using the number of stages for process optimization is best applied if either mass transfer or dispersion dominates the peak broadening. Therefore, the optimization strategies discussed later in this chapter apply a validated transport dispersive model, which can flexibly consider mass transfer and/ or dispersion effect. Here, the number of stages is used as independent variable for the optimization criteria like productivity or eluent consumption. Another possible approach would be the use of simplified simulation model like equilibrium dispersive model (Seidel-Morgenstern, 1995). 

No matter what the price strategy of firm Y, a firm X price of 80 cents will capture more market share than any other X price. Similarly, firm Y will come to the same conclusion and choose 80 cents. The two firms will share the market equally. This is an example of a zero-sum game, where the two finns compete and the gain of one firm is the loss of the other. An underlying assumption is that of constant product demand. The solution of this game results in a pure strategy dominated by a price of 80 cents for both finns. This is a stable-equilibrium situation that is, neither firm can improve its market position by making unilateral moves. 

The most studied flow type is simple shear flow, which is dominant in all pressure- and drag-driven flows. The strategies adopted, after keeping the material for a sufficient period above the equilibrium melting point in order to erase any history effects, are ... 

---