Uncertainty Aversion with Second-Order Probabilities and Utilities
Robert F. Nau
Fuqua School of Business
Duke University
https://www.doczj.com/doc/4516952016.html,/~rnau
RUD 2002 version
May 21, 2002
Abstract
Uncertainty aversion is often conceived as a local, first-order effect that is associated with kinked indifference curves (i.e., non-smooth preferences), as in the Choquet expected utility model. This paper shows that uncertainty uncertainty aversion can arise as a global, second-order effect when the decision maker has smooth preferences within the state-preference framework of choice under uncertainty, as opposed to the Savage or Anscombe-Aumann frameworks. Uncertainty aversion is defined and measured in direct behavioral terms without reference to probabilistic beliefs or consequences with state-independent utility. A simple axiomatic model of “partially separable” non-expected utility preferences is presented, in which the decision maker satisfies the independence axiom selectively within partitions of the state space whose elements have similar degrees of uncertainty. As such, she may behave like an expected-utility maximizer with respect to assets in the same uncertainty class, while exhibiting higher degrees of risk aversion toward assets that are more uncertain. An alternative interpretation of the same model is that the decision maker may be uncertain about her credal state (represented by second-order probabilities for her first-order probabilities and utilities), and she may be averse to that uncertainty (represented by a second-order utility function). The Ellsberg and Allais paradoxes are explained by way of illustration. Keywords: risk aversion, uncertainty aversion, ambiguity, non-additive probabilities, state-preference theory, Choquet expected utility, Ellsberg’s paradox.
1. Introduction
The axiomatization of expected utility by von Neumman-Morgenstern and Savage hinges on the axiom of independence, which requires preferences to be separable across mutually exclusive events and leads to representations by utility functions that are additively separable across states of the world. A strong implication of the independence axiom is that preferences among acts do not depend on qualitative properties of events but rather only on the sum of the values attached to their constituent states, which in turn depend only on the probabilities of the states and the consequences to which they lead. If the set of states of nature can be partitioned in two or more ways, the decision maker is not permitted to display uniformly different risk attitudes toward acts that are measurable with respect to different partitions, because the events in any two partitions are composed of the same states, merely grouped in different ways. There is considerable empirical evidence that individuals violate this requirement in certain kinds of choice situations. One example is provided by Ellsberg’s 2-color paradox, in which subjects consistently prefer to bet on unambiguous events rather than ambiguous events, even when they are otherwise equivalent by virtue of symmetry, a phenomenon known as uncertainty aversion. Other violations of independence are provided by Ellsberg’s 3-color paradox and Allais’ paradox, in which subjects’ preferences between two acts that agree in some events depend on how they agree there, as though there were complementarities among events.
A variety of models of non-expected utility have been proposed to accommodate violations of the independence axiom, and most of them do so by positing that the decision maker has non-probabilistic beliefs or that her preferences depend nonlinearly on probabilities. The Choquet expected utility model, in particular, assumes that the decision maker tends to overweight events leading to inferior payoffs by applying non-additive subjective probabilities derived from a Choquet capacity. (Schmeidler 1989) The ranking of
states according to the payoffs to which they lead thus plays a key role in the representation of uncertainty aversion: the decision maker violates the axioms of expected utility theory only when faced with choices among acts whose payoffs induce different rankings of states. Within sets of acts whose payoffs are comonotonic, the decision maker’s preferences have an ordinary expected-utility representation. Schmeidler derives the Choquet model within the Anscombe-Aumann framework of choice under uncertainty, which includes objective probabilities in the composition of acts, and he defines uncertainty aversion in this framework as convexity with respect to mixtures of probabilities. That is, an uncertainty-averse individual who prefers f to g also prefers αf+ (1?α)g to g for 0<α<1,where the mixture operation applies to the objective probabilities with which acts f and g lead to various consequences. Another way to view the Choquet model is to note that it implies that the decision maker has indifference curves in payoff space that are kinked in particular locations, namely at the boundaries between comonotonic sets of acts. If the decision maker’s status quo wealth happens to fall on such a kink—which is a set of measure zero—her local preferences (i.e., her preferences for “neighboring” acts) will display uncertainty aversion, otherwise they will not.
Epstein (1999) takes a different approach, in which uncertainty aversion is defined as a departure from probabilistically sophisticated behavior (Machina and Schmeidler 1992). A probabilistically sophisticated decision maker may have non-expected-utility preferences but nevertheless has an additive subjective probability measure that is uniquely determined by preferences. Epstein (following Machina and Schmeidler) uses the Savage-act modeling framework, in which objects of choice are mappings from states to consequences whose utilities are a priori state-independent. There are no objectively mixed acts; rather, Epstein’s definition of uncertainty aversion assumes the existence of a set of special acts that are a priori unambiguous. A preference order 1! is defined to be more uncertainty averse than
preference order 2! if, for all acts w and all unambiguous acts y, y1!w(y1!w) whenever y2!w( y 2!w). In other words, order 1 is more uncertainty averse than order 2 if 1 never chooses an ambiguous act over an unambiguous act when 2 would not. A preference order is then defined to be uncertainty averse if it is more uncertainty averse than some probabilistically sophisticated preference order. Epstein and Zhang (2001) extend the same approach to define the set of unambiguous events endogenously: an unambiguous event is one that does not give rise to preference reversals when a common consequence assigned to that event is replaced by another common consequence.
Ghirardato and Marinacci (2001, 2002) give a definition of uncertainty aversion that, like Epstein’s, is based on a definition of comparative uncertainty aversion but is more general in the sense that it does not assume the existence of acts that are a priori unambiguous, aside from the constant acts that play a key role in Savage’s framework. However, Ghirardato and Marinacci’s definition imposes the additional restriction that preferences should be biseparable—i.e., that when choices are restricted to binary gambles, preferences should be represented by a utility function of the form V(xAy) =u(x)ρ(A) + u(y)(1?ρ(A)), where u is a cardinal utility function for money and ρ is a “willingness to bet”measure. Uncertainty aversion is exhibited when ρ is nonadditive, i.e., ρ(A)+ρ(A) < 1.
The preceding definitions of uncertainty aversion are motivated by the Choquet expected utility model and related models in which uncertainty aversion is associated with kinked indifference curves—thus, uncertainty aversion is conceived as a first-order rather than second-order effect (Segal and Spivak 1990)—and they all require a priori definitions of riskless (constant) acts in the tradition of Savage and Anscombe-Aumann. The aim of the present paper is to consider, instead, situations in which (i) there are no a priori riskless acts or consequences whose utilities are state-independent, (ii) the decision maker has smooth preferences so that both risk aversion and uncertainty aversion are second-order effects, and
(iii) the decision maker may be locally uncertainty averse at all wealth positions, not merely at “special” wealth positions. The definition of uncertainty aversion given here is somewhat similar in spirit to that of Ghirardato and Marinacci, insofar a decision maker is considered uncertainty averse if she is “more uncertainty averse” than a decision maker with separable preferences. However, unlike Schmeidler, Epstein, and Ghirardato/Marinacci, we make no attempt to uniquely separate beliefs from state-dependent cardinal utilities, on the grounds that such a separation is impossible to achieve in practice and is not necessary for purposes of economic modeling or decision analysis (Nau 2002). Roughly speaking, a decision maker will be defined here to be uncertainty averse if she is systematically more risk averse toward acts that are “more uncertain” in a manner that cannot be explained by a state-dependent Pratt-Arrow measure of risk aversion. For example, she might be risk neutral—or even locally risk seeking—toward casino gambles, moderately risk averse toward investments in the stock market, and highly risk averse when insuring against health or property risks.
The organization of the paper is as follows. Section 2 introduces the basic mathematical framework, and section 3 defines risk aversion and uncertainty aversion for a decision maker with smooth preferences. Section 4 gives a simple example of a utility function defined on a 4-element state space that rationalizes Ellsberg’s 2-color paradox. Section 5 presents a more detailed and general version of the same model and the axioms of “partially separable preferences” on which it rests. Section 6 presents an alternative interpretation of the model in terms of second-order uncertainty about credal states, illustrated by a model of Ellsberg’s 3-color paradox as well as the 2-color paradox. Section 7 presents two different models for the Allais paradox, one that is based on second-order uncertainty about probabilities and another that is based on second-order uncertainty about utilities. Section 8 discusses how the preference model differs from Choquet expected utility, and Section 9 presents some concluding comments.
2. Preliminaries
The modeling framework used throughout this paper is that of state-preference theory (Arrow 1953/1964, Debreu 1959, Hirshleifer 1965), which encompasses both expected-utility and non-expected-utility models of choice under uncertainty.1 Suppose that there are n mutually exclusive, collectively exhaustive, states of nature and a single divisible commodity (money) in terms of which payoffs are measured. The wealth distribution of a decision maker is represented by a vector w ∈ ?n , whose j th element w j denotes the quantity of money received in state j , in addition to (unobserved) status quo wealth.
Assumption 1: The decision maker’s preferences among wealth distributions satisfy the usual axioms of consumer theory (reflexivity, completeness, transitivity,continuity, and monotonicity), as well as a smoothness property, so that they are representable by a twice-differentiable ordinal utility function U (w ) that is a non-decreasing function of wealth in every state.
By the monotonicity assumption, the gradient of U at wealth w is a non-negative vector that can be normalized to yield a probability distribution π(w ) whose j th element is
∑=????=n i i j
j w U w U 1)/()/()(w w w π.
Thus, πj (w ) is the rate at which the decision maker would indifferently bet infinitesimal amounts of money on or against the occurrence of state j .2 π(w ) is invariant to monotonic
1 For our purposes, the appeal of the state-preference framework is that it uses concrete sums of money as outcomes without necessarily assuming that utility for money is state-independent or that background wealth is observable or that utilities and subjective probabilities are uniquely separable.
2 Because preferences are assumed to be both complete and smooth, we rule out first-order aversion to risk or uncertainty, in which, even for infinitesimal stakes, there might be no rate at which the decision maker would indifferently bet on or against a given event.
transformations of U and is observable, regardless of whether the decision maker is probabilistically sophisticated. It is commonly known as a risk neutral probability distribution because the decision maker prices very small assets in a seemingly risk-neutral manner with respect to it. A state will be defined here to be non-null if its risk neutral probability is positive at every wealth distribution. A decision maker’s risk neutral probabilities at wealth w are implicitly a function of her beliefs, her attitudes toward risk and uncertainty, and her personal and financial stakes in events, although their effects are often confounded.
The risk-neutral distribution π(w ) describes the first-order properties of the decision maker’s local preferences in the vicinity of wealth w . Second-order properties of local preferences are described by the matrix R (w ) whose jk th element is
j k
j jk w U w w U r ??????=)/()/()(2w w w .
R (w ) generalizes the familiar Pratt-Arrow measure to the present setting and will be called the risk aversion matrix . Hence, the decision maker’s local preferences at wealth w are described, up to second-order effects, by a risk neutral probability distribution π(w ) and a risk aversion matrix R (w ).3
If no additional restrictions are placed on preferences, the individual is rational by the standard of no-arbitrage, but she may have non-expected utility preferences in the sense that
3 The matrix R (w ) is not observable in the general case, just as the ordinal utility function U is not observable.U is determined by preferences only up to monotonic transformations, and correspondingly R (w ) is uniquely determined by preferences only up to the addition of constants to each column. However, R (w ) can be
normalized subtracting a constant from each column so that the risk neutral expectation of each column is zero,and the normalized matrix is observable. (Nau 2001) Here the “unnormalized” matrix R (w ) will be used for convenience, but the results could as well be recast in terms of the normalized matrix.
her valuation of a risky asset may not be decomposable into a product of probabilities for states and utilities for consequences. For example, she may behave as if amounts of money received in different states are substitutes or complements for each other, which is forbidden under expected utility theory. However, if her preferences are additionally assumed to satisfy the independence axiom (Savage’s P2), then U(w) has an additively separable representation: U(w) = v1(w1) + v2(w2) + … + v n(w n).
In this case, R(w) reduces to an observable diagonal matrix. If preferences are further assumed to be conditionally state-independent (Savage’s P3), then U(w) has a state-independent expected-utility representation:
U(w) = p1u(w1) + p2u(w2) + … + p n u(w n),
where p is a unique probability distribution and u(x) is a state-independent utility function that is unique up to positive affine scaling, as in Savage’s model.4 Although the probabilities in the latter representation are unique, it does not yet follow that they are the decision maker’s “true” subjective probabilities, because there are many other equivalent representations in which different probabilities are combined with state-dependent utilities. In order for the decision maker to be probabilistically sophisticated—i.e., in order for her preferences to determine a unique ordering of events by probability—an additional qualitative probability axiom (Savage’s P4 or Machina and Schmeidler’s P4*, 1992) is needed, together with an a priori definition of consequences whose utility is “constant”
4 For a decision maker who is a state-independent expected-utility maximizer, risk neutral probabilities are merely the product of true subjective probabilities and relative marginal utilities for money at the current wealth position, i.e., πj(w) ∝p j u′(w j), where p j is the subjective probability of state j and u(w) is the state-independent utility of wealth w. If an attempt is made to elicit the decision maker’s subjective probabilities by de Finetti’s method—i.e., by asking which gambles she is willing to accept—the probabilities that are observed will be her risk neutral probabilities rather than her true probabilities.
across states of nature. However, preference measurements among feasible acts do not reveal whether the consequences are, in fact, constant, and so the unique separation of probability from cardinal utility remains problematic and controversial (Schervish et al. 1990, Karni and Mongin 2000, Karni and Grant 2001, Nau 2002, Karni 2002). For this reason, it is of interest to try to define both risk aversion and uncertainty aversion in a way that does not depend on the a priori identification of a set of constant consequences nor on the assumption of state-independent preferences.5
3. Definitions of aversion to risk and uncertainty
In order to characterize aversion to uncertainty, it is necessary to begin with a characterization of aversion to risk. Following Yaari (1969) the decision maker is defined to be risk averse if her preferences are convex with respect to mixtures of payoffs, which means that whenever w is preferred to w*, αw+(1?α)w* is also preferred to w* for 0<α<1, where αw+(1?α)w*denotes the wealth distribution whose monetary value in state i is αw i+(1?α)w i*.6 Payoff-convexity of preferences implies that the ordinal utility function U is quasi-concave. This definition of risk aversion does not require prior definitions of expected value or absence of risk.7 The decision maker’s degree of local risk aversion can be
5 Epstein and Zhang (2001) explicitly invoke Savage’s state-independent utility axiom (P3) and observe: “The necessity of this axiom in our approach implies, in particular, that we have nothing to say about the meaning of ambiguity when preferences are state-dependent.” [p. 275, emphasis added]
6 Note that there is a suggestive duality between Yaari’s definition of risk aversion as payoff-convexity and Schmeidler’s definition of uncertainty aversion as probability-convexity. However, Schmeidler’s definition is criticized by Epstein (1999) and cannot be applied in the present framework, which lacks objective probabilities.
7 See Machina (1995) and Karni (1995) for a discussion of the implications of different definitions of risk aversion under non-expected utility preferences.
measured by the difference between her risk-neutral valuation of an asset and the price at which she is actually willing to buy it. Let z denote the payoff vector of a risky asset. The decision maker’s buying price for z at wealth w, denoted B(z;w), is determined by U(w+z–B(z;w)) ?U(w) = 0.
The (buying) risk premium associated with z at wealth w,here denoted b(z;w), is the difference between the asset’s risk neutral expected value and its buying price: b(z;w) = Eπ(w)[z] – B(z;w).
It follows (as a consequence of quasi-concavity) that the decision maker is risk averse if and only if her risk premium is non-negative for every asset z at every wealth distribution w.
If z is a neutral asset (Eπ(w)[z] = 0), its risk premium has the following second-order approximation that generalizes the Pratt-Arrow formula (Nau 2001):
b(z;w) ≈ ? z T Π(w)R(w) z,
where Π(w)= diag(π(w)). In the special case where U is additively separable (i.e., the decision maker satisfies the independence axiom), R(w)is an observable diagonal matrix and the risk premium formula reduces to
b(z;w) ≈ ? Eπ(w)[r(w)z2],
where r(w) is a vector-valued risk aversion measure whose j th element is
r j(w) = ?U jj(w)/U j(w).
Hence, for a decision maker with quasi-concave additively-separable utility, local preferences are completely described (up to second-order effects) by a pair of numbers for each state: a risk neutral probability πj(w) and a risk aversion coefficient r j(w). Such a decision maker is risk averse but uncertainty neutral, inasmuch as her preferences have an expected-utility representation even if her probabilities cannot be uniquely separated from her (possibly state-dependent) utilities.
It remains to characterize aversion to uncertainty in a simple behavioral way. For
convenience, in the spirit of Epstein (1999), we assume that there is a reference set of events that are a priori “unambiguous.”8
Assumption 2: There is a set of unambiguous events, closed under complementation and disjoint union, at least one of which is a union of two or more non-null states and whose complement is also a union of at two or more non-null states.
Intuitively, a decision maker is averse to uncertainty if she prefers to bet on unambiguous rather than ambiguous events, other things being equal. To make this notion precise, we introduce the notion of an A:B ?-spread (“A vs. B delta-spread”). Let A and B denote two logically independent events, let {AB π,B A π,B A π,B A π}denote the local risk neutral probabilities (at wealth w ) of the four possible joint outcomes of A and B , and assume they are all strictly positive. Let ? denote a quantity of money (just) large enough in magnitude that second-order utility effects are relevant. Then an A:B ?-spread and a B:A ?-spread are defined as the neutral assets whose payoffs are given by the following two contingency tables:
B B A
A A A Table 1: ?-spreads defined
8 The unambiguous events play no essential role in the analysis except to distinguish whether a particular pattern of behavior is ambiguity-averse or ambiguity-loving. Since unambiguous events can be created if necessary by flipping coins, this entails no great loss of generality. However, we do not assume that unambiguous events necessarily have have objective or otherwise uniquely determined probabilities.
Note that an A:B ?-spread is a non-simple bet on A: the sign of the payoff depends only on whether A occurs or not, but the magnitude may also depend on B. Similarly, a B:A ?-spread is a non-simple bet on B whose payoff magnitude may depend on A. In both cases, the winning and losing amounts are scaled so that the contribution to the risk neutral expected payoff is ±? in every cell of the table. By construction, both bets are neutral and both must have the same risk premium for a decision maker with separable preferences, because for such a person the risk premium is obtained (to a second order approximation) by squaring the payoffs, multiplying them by the statewise risk neutral probabilities and risk aversion coefficients, and summing.
Definition 1: The decision maker is locally uncertainty averse at wealth w if, for
every unambiguous event B, every event A that is logically independent of B, and any ? sufficiently small in magnitude (positive or negative), a B:A ?-spread is weakly
preferred to (i.e., has a risk premium less than or equal to that of) an A:B ?-spread.
The decision maker is uncertainty averse if she is locally uncertainty averse at every wealth position.
The classic demonstration of uncertainty aversion is provided by Ellsberg’s 2-color paradox, in which a subject prefers to stake a modest prize (say, $400) on the draw of a ball from an urn with equal proportions of red and black balls rather than a draw from an urn with unknown proportions of red and black, regardless of the winning color. To adapt this experiment to the framework of Definition 1, let B denote the unambiguous event that a red ball is drawn from urn 1 (with known 50-50 proportions of red and black) and let A denote the potentially ambiguous event that a red ball is drawn from urn 2 (with unknown proportions of red and black). By symmetry, if there are no prior stakes, it is reasonable to suppose that the decision maker’s risk neutral probabilities for the four possible joint outcomes are identically 1/4. Then, according to Definition 1, an uncertainty averse decision
maker would prefer to receive +4? in the event that red is drawn from urn 1 and ?4? in the event that black is drawn from urn 1 (a B:A ?-spread ) rather receive +4? in the event that red is drawn from urn 2 and ?4? in the event that black is drawn from urn 2 (an A:B ?-spread), regardless of whether ? is +$100 or ?$100.
In Ellsberg’s three-color paradox, there is a single urn containing 30 red balls and 60 balls that are black and yellow in unknown proportions. A typical subject prefers to stake (say) a $300 prize on the draw of a red ball rather than a black ball when $0 is to be received in either case if a yellow ball is drawn, but the direction of preference is reversed when $300 is to be received if yellow is drawn. The 3-color paradox is more transparently a violation of the independence axiom but less transparently an example of aversion to uncertainty. It is usually presented as a set of choices among prospects that involve only gains:
a b c d e f
Red300000300300
Black030003000300
Yellow003003003000
Table 2: Ellsberg’s 3-color paradox in gains-only form
The typical response pattern is a > b ~ c and d > e ~ f. When the prospects are converted to “fair” gambles by subtraction of appropriate constants, the choices look like this:
a′b′c′d′e′f′
Red200-100-100-200100100
Black-100200-100100-200100
Yellow-100-100200100100-200
Table 3: Ellsberg’s 3-color paradox in fair-gamble form
The last three gambles are merely the opposite sides of the first three. We would expect an uncertainty-averse subject to give the same pattern of responses, i.e., a′ > b′ ~ c′ and d′ > e′ ~ f′, and indeed the Choquet expected utility model predicts exactly this result. However, a′ >
b′ ~ c′ and d′ > e′ ~ f′ does not violate the independence axiom. For example, it is consistent with a model of state-dependent expected-utility preferences of the following form: U(w) = ?( ?2 exp(?? αw1)) + ?(?exp(?αw2)) + ?(?exp(?αw3)),
where w1, w2, w3 are the amounts of wealth received under Red, Black, and Yellow, respectively, and α is any positive number. Under this utility function, the risk neutral probabilities of the three events are equal when w1= w2= w3 = 0, but the Pratt-Arrow measure of risk aversion in the event Red is only half as large as it is in the other two events, hence the decision maker prefers fair gambles in which the largest change in wealth occurs under Red, regardless of whether it is positive or negative. For example, with α= 0.001, the risk premium is 6.667 for both a′ and d′ , while it is 9.167 for the other prospects.9 To embed the 3-color paradox in the framework of Definition 1, which requires the state space to consist of at least four elements and to be partioned by at least one unambiguous event, it is necessary to create a fourth state. Therefore, consider the following extension of the 3-color experiment: a fair coin is flipped, and (only) if tails is obtained, a ball is then drawn from the urn. The four possible outcomes are then {Red, Black, Yellow, Heads}, where Red and Heads (and their union) are unambiguous. By symmetry, if there are no prior stakes, it is reasonable to assume that the corresponding risk neutral probabilities are (1/6, 1/6, 1/6, 1/2). Consider the following two ?-spreads:
9 This example illustrates the complications that state-dependent preferences create for some definitions of ambiguity, alluded to by Epstein and Zhang (2001).
Red∪Heads:Black∪Heads
?-spread Black∪Heads:Red∪Heads
?-spread
Red+6??6?
Black?6?+6?
Yellow?6??6?
Heads+2?+2?
Table 4: ?-spreads for the extended 3-color paradox
Thus, the decision maker receives +2? in any case if Heads occurs. If Heads does not occur, the first column is a bet on or against Red, while the second is a bet on or against Black. (The direction of the bet depends on the sign of ?.) Because Red∪Heads is unambiguous, an uncertainty-averse decision maker prefers the first bet over the second, regardless of whether ? is (say) +$50 or ?$50.
The preceding examples motivate a definition of comparative ambiguity between events:
Definition 2: (a) If A1 and A2 are logically independent and their four joint
outcomes are non-null, then an uncertainty averse decision maker regards A1 as less
ambiguous than A2 if for any ? sufficiently small in magnitude (positive or negative), an A1:A2 ?-spread is strictly preferred to an A2:A1 ?-spread at every wealth
distribution.
(b) If A1 and A2 are disjoint, and the state space can be partioned as {A1, A2, A3, B}
where all four events are non-null and B is unambiguous, then an uncertainty averse
decision maker regards A1 as less ambiguous than A2 if, for any ? sufficiently small in magnitude (positive or negative), an A1∪B:A2∪B?-spread is strictly preferred to an
A2∪B:A1∪B?-spread at every wealth distribution.
4. A simple model of smooth preferences explaining the Ellsberg paradox
The definitions of the preceding section provide simple tests for presence of second-order uncertainty aversion and for comparing the degree of ambiguity of different events. We now present an example of a utility function that exhibits second-order uncertainty aversion in Ellsberg’s 2-color paradox. Henceforth, let A1 [A2] denote the event that the ball drawn from the unknown urn is red [black], and let B1[B2] denote the event that the ball drawn from the known urn is red [black]. The relevant state space is then {A1B1, A1B2, A2B1, A2B2}. Unless the subject has a strict color preference and/or prior stakes in the outcomes of the events (which we assume she does not), the four states are completely symmetric when considered one-at-a-time: each is the conjunction of an ambiguous event (A1 or A2) and an unambiguous event (B1 or B2) differing only in their color associations. If the subject had to choose one of the four states on which to stake a prize, she would have no basis for a strict preference. The paradox lies in the fact that the states are not symmetric when considered two-at-a-time: the pair of states{A1B1, A2B1} has an objectively known probability while the pair of states {A1B1, A1B2} does not.
Let w = (w11, w12, w21, w22) denote the decision maker’s wealth distribution, where w ij is wealth in state A i B j, and suppose that she evaluates wealth distributions according to the following non-separable utility function:
(1)U(w) = –p1 exp(–α(q11 w11 + q12 w12)) – p2exp(–α(q21 w21 + q22w22))
where α is a positive constant and p1 = p2 = q11 = q12 = q21 = q22 = ?. It is natural to interpret p i as a marginal probability for A i and q ij as a conditional probability of B j given A i. Suppose that the prior wealth distribution is an arbitrary constant—i.e., the subject has no prior stakes in the draws from either urn. Then the states are symmetric with respect to changes in wealth one-state-at-a-time, from which it follows that π(w) = (?, ?, ?, ?). Hence, for infinitely small bets, the subject does not distinguish among the states, exactly as if she were a state-
independent expected utility maximizer with uniform prior probabilities and no prior stakes. However, she does distinguish among the states when considering bets in which states are grouped together and in which the stakes are large enough for risk aversion to come into play. For example,the subject would prefer to pair a finite gain in state A1B1 with an equal loss in state A1B2 (yielding no change in U) rather than with an equal loss in state A2B1 or state A2B2 (yielding a decrease in U). The corresponding risk aversion matrix is R(w) ≡ ?αC, where C has a block structure with 1’s in its upper left and lower right 2×2 submatrices.
Now consider the following three neutral bets. In bet #1, the subject wins x>0 if a red ball is drawn from the known urn and loses x otherwise, so that the payoff vector is (x, ?x, x,?x), i.e., a B:A(x/4)-spread. In bet #2, the subject wins x if a red ball is drawn from the unknown urn and loses x otherwise, so that the payoff vector is (x, x, ?x, ?x), i.e., an A:B (x/4)-spread. In bet #3, the subject wins x if the balls drawn from urns 1 and 2 are the same color, and loses x otherwise, so that the payoff vector is (x, ?x, ?x, x). Applying the formula b(z; w) = ? z TΠ(w)R(w)z, the risk premium for bet #1 is zero, and the same is true if the sign of x is reversed so that black is the winning color. Hence, the subject is risk neutral with respect to bets on the known urn. Whereas, the risk premium for bet #2 is ?αx2, and the same risk premium is obtained if the winning color is changed to black. Hence, the subject is risk averse with respect to bets on the unknown urn:she behaves toward it as if she believes red and black are equally likely but her Pratt-Arrow risk aversion coefficient is equal to α. This pattern of risk neutral behavior toward unambiguous events and risk averse behavior toward ambiguous events exemplifies uncertainty aversion as given in Definition 1. Interestingly, the risk premium for bet #3 is the same as for bet #1, namely zero, and the same is true if the sign of x is reversed so that the subject wins if the balls are of different colors. Hence, the subject behaves risk neutrally with respect to the unknown urn when the winning
color is determined by “objective” randomization using the known urn, which is a well-known trick for eliminating the ambiguity.
5. A general model of partially separable preferences
The example in the preceding section suggests a novel hypothesis about the character of non-expected-utility preferences, namely that the decision maker may behave like an expected-utility maximizer with respect to assets in the same “uncertainty class,” while exhibiting higher degrees of second-order risk aversion toward assets that are “more uncertain.” This section presents an axiomatic model of such preferences. Henceforth, let the state space consist of a Cartesian product A×B, where A ={A1, …, A m} and B = {B1, …, B n} are finite partitions, and A-measurable events are potentially ambiguous while B-measurable events are a priori unambiguous. To fix ideas, it might be supposed that the elements of A are “natural” states of the world, while elements of B are states of an artificial randomization device. (Another interpretation will be suggested later.)
Let w, w*, z, z*, denote wealth distributions over states, i.e., monetary acts. For any event E and acts w and z, let E w + (1?E)z denote the act that agrees with w on E and agrees with z on E. Suppose that preferences among acts satisfy the following partition-specific independence axioms.
Assumption 3:
A-independence: E w + (1?E)z≥E w* + (1?E)z?E w + (1?E)z*≥E w* + (1?E)z*
for all acts w, w*, z, z* and every A-measurable event E, and conditional preference
w≥E w*is accordingly defined for such events.
B -independence: F w + (1?F )z ≥i F w * + (1?F )z ? F w + (1?F )z * ≥i F w * + (1?F )z *for every B -measurable event F , where ≥i denotes conditional preference given
element A i of A .
In other words, the decision maker satisfies the independence axiom unconditionally with respect to A -measurable events and conditionally with respect to B -measurable acts and events within each element of A . Such a person will be said to have partially separable preferences. A -independence and B -independence are similar to the time-0 and time-1substitution axioms of Kreps and Porteus (1979), adapted to a framework of choice under uncertainty rather than risk and stripped of their temporal interpretation. Our main result is the following
PROPOSITION:
(a) Under Assumptions 1?3, preferences are represented by a utility function U
having the composite-additive form:
(2)U (w ) = ))
((11ij n
j ij m i i w v u ∑∑==where w ij denotes wealth in state A i B j , and {u i } and {v ij } are non-decreasing twice-differentiable state-dependent utility functions.
(b) The corresponding risk neutral probabilities satisfy
πij = )())(()
())((1111
hl hl hk n k hk n l h m
h ij ij ik n k ik i w v w v u w v w v u ′′′′∑∑∑∑==== .
(c) The local risk aversion matrix R (w ) is the sum of a diagonal matrix and a block-diagonal matrix, with r ij ,kl (w ) = 0 if i ≠k and
jl
ij ij ij ij il il ih n h ih i ih n h ih i il ij w v w v w v w v u w v u r 1))(/)(()))
(())
(()((11,′″+′′″=
∑∑==w (d) Let s and t denote m - and mn-vectors defined by:
s i = ∑∑∑===′′″
n
h ih ih ih n
h ih i ih n h ih i w v w v u w v u 111
)())(())
((t ij = v j ″(w ij )/v j ′(w ij ).
Let πi = ∑=n
h ih 1πand πh |i = πih /πi denote the induced marginal and conditional probabilities, and let z be defined as the m -vector whose i th element is ih m
h i h i z z ∑==1|π,
i.e., the conditional risk-neutral expectation of z given event A i . In these terms, the
risk premium for a neutral asset z is
b (z ; w ) ≈ 21121π?π?ij ij m i n j ij i
m i i i z t z s ∑∑∑===+ = ? E π[s z 2] + ? E π[t z 2].
(e) The decision maker is locally uncertainty averse if s ≥ 0 and globally uncertainty averse if u i is concave for every i .
Proof: Part (a) follows from a double application of the usual argument showing that the independence axiom leads to an additively separable utility representation.
(Debreu 1960, Wakker 1989) Parts (b)-(d) are obtained by some algebra and the risk premium formula from Nau (2001). For part (e), note that the off-diagonal terms of R (w ) are non-negative if s ≥ 0, and this is true at all w if u i is concave for every i . If A is any A -measurable (potentially ambiguous) event and B is any B -measurable
(unambiguous) event, the risk premium of an A:B ?-spread must be greater than or
英语介词用法大全 TTA standardization office【TTA 5AB- TTAK 08- TTA 2C】
介词（The Preposition）又叫做前置词，通常置于名词之前。它是一种虚词，不需要重读，在句中不单独作任何句子成分，只表示其后的名词或相当于名词的词语与其他句子成分的关系。中国学生在使用英语进行书面或口头表达时，往往会出现遗漏介词或误用介词的错误，因此各类考试语法的结构部分均有这方面的测试内容。 1. 介词的种类 英语中最常用的介词，按照不同的分类标准可分为以下几类： (1). 简单介词、复合介词和短语介词 ①.简单介词是指单一介词。如： at , in ,of ,by , about , for, from , except , since, near, with 等。②. 复合介词是指由两个简单介词组成的介词。如： Inside, outside , onto, into , throughout, without , as to as for , unpon, except for 等。 ③. 短语介词是指由短语构成的介词。如： In front of , by means o f, on behalf of, in spite of , by way of , in favor of , in regard to 等。 (2). 按词义分类 {1} 表地点（包括动向）的介词。如： About ,above, across, after, along , among, around , at, before, behind, below, beneath, beside, between , beyond ,by, down, from, in, into , near, off, on, over, through, throught, to, towards,, under, up, unpon, with, within , without 等。 {2} 表时间的介词。如： About, after, around , as , at, before , behind , between , by, during, for, from, in, into, of, on, over, past, since, through, throughout, till(until) , to, towards , within 等。 {3} 表除去的介词。如： beside , but, except等。 {4} 表比较的介词。如： As, like, above, over等。 {5} 表反对的介词。如： againt ,with 等。 {6} 表原因、目的的介词。如： for, with, from 等。 {7} 表结果的介词。如： to, with , without 等。 {8} 表手段、方式的介词。如： by, in ,with 等。 {9} 表所属的介词。如： of , with 等。 {10} 表条件的介词。如：
介词用法知多少 介词是英语中最活跃的词类之一。同一个汉语词汇在英语中可译成不同的英语介词。例如汉语中的“用”可译成：（1）用英语（in English）；（2）用小刀（with a knife）；（3）用手工（by hand）；（4）用墨水（in ink）等。所以，千万不要以为记住介词的一两种意思就掌握了这个介词的用法，其实介词的用法非常广泛，搭配能力很强，越是常用的介词，其含义越多。下面就简单介绍几组近义介词的用法及其搭配方法。 一. in, to, on和off在方位名词前的区别 1. in表示A地在B地范围之内。如： Taiwan is in the southeast of China. 2. to表示A地在B地范围之外，即二者之间有距离间隔。如： Japan lies to the east of China. 3. on表示A地与B地接壤、毗邻。如： North Korea is on the east of China. 4. off表示“离……一些距离或离……不远的海上”。如： They arrived at a house off the main road. New Zealand lies off the eastern coast of Australia. 二. at, in, on, by和through在表示时间上的区别 1. at指时间表示： （1）时间的一点、时刻等。如： They came home at sunrise （at noon, at midnight, at ten o’clock, at daybreak, at dawn）. （2）较短暂的一段时间。可指某个节日或被认为是一年中标志大事的日子。如： He went home at Christmas （at New Year, at the Spring Festival, at night）. 2. in指时间表示： （1）在某个较长的时间（如世纪、朝代、年、月、季节以及泛指的上午、下午或傍晚等）内。如： in 2004, in March, in spring, in the morning, in the evening, etc （2）在一段时间之后。一般情况下，用于将来时，谓语动词为瞬间动词，意为“在……以后”。如： He will arrive in two hours. 谓语动词为延续性动词时，in意为“在……以内”。如： These products will be produced in a month. 注意：after用于将来时间也指一段时间之后，但其后的时间是“一点”，而不是“一段”。如： He will arrive after two o’clock. 3. on指时间表示： （1）具体的时日和一个特定的时间，如某日、某节日、星期几等。如： On Christmas Day（On May 4th）, there will be a celebration. （2）在某个特定的早晨、下午或晚上。如： He arrived at 10 o’clock on the night of the 5th. （3）准时，按时。如： If the train should be on time, I should reach home before dark. 4. by指时间表示： （1）不迟于，在（某时）前。如：
with用法归纳 （1）“用……”表示使用工具，手段等。例如： ①We can walk with our legs and feet. 我们用腿脚行走。 ②He writes with a pencil. 他用铅笔写。 （2）“和……在一起”，表示伴随。例如： ①Can you go to a movie with me? 你能和我一起去看电影'>电影吗？ ②He often goes to the library with Jenny. 他常和詹妮一起去图书馆。 （3）“与……”。例如： I’d like to have a talk with you. 我很想和你说句话。 （4）“关于，对于”，表示一种关系或适应范围。例如： What’s wrong with your watch? 你的手表怎么了？ （5）“带有，具有”。例如： ①He’s a tall kid with short hair. 他是个长着一头短发的高个子小孩。 ②They have no money with them. 他们没带钱。 （6）“在……方面”。例如： Kate helps me with my English. 凯特帮我学英语。 （7）“随着，与……同时”。例如： With these words, he left the room. 说完这些话，他离开了房间。 [解题过程] with结构也称为with复合结构。是由with+复合宾语组成。常在句中做状语，表示谓语动作发生的伴随情况、时间、原因、方式等。其构成有下列几种情形： 1.with+名词(或代词)+现在分词 此时，现在分词和前面的名词或代词是逻辑上的主谓关系。 例如:1)With prices going up so fast, we can't afford luxuries. 由于物价上涨很快，我们买不起高档商品。（原因状语） 2)With the crowds cheering, they drove to the palace. 在人群的欢呼声中，他们驱车来到皇宫。（伴随情况） 2.with+名词(或代词)+过去分词 此时，过去分词和前面的名词或代词是逻辑上的动宾关系。
高中三年自我鉴定范文 高中三年,痛并快乐着,累却充实着,苦但愉悦着,要毕业了为这高中三年做义工自我鉴定，本文是WTT小雅为大家整理的高中三年自我鉴定范文篇，仅供参考。 高中三年自我鉴定范文篇一 蓦然回首，高中学习生活即将结束。回想三年里，父母的殷切期望，老师的谆谆教导，同学间的友爱相助。曾经付出的辛勤汗水，曾经的收获，曾经的幸福快乐。曾经少不更事懵懵懂懂的我，一幕幕似在眼前，令人感慨万千。今天的我，德智体美劳全面发展，从幼稚一步步走向了成熟。 我立场坚定正确，热爱祖国，热爱党。我自觉践行社会主义核心价值观，积极要求进步，踊跃参加学校各种社团组织。高一即通过竞选参加青州二中学生会，并成为纪检部部长，能够在学校领导安排协调下，团结学生会其他成员和各班同学，积极主动地开展学校日常纪律执勤，大型会议赛事秩序维护，组织安排等事务性工作。同时我也积极参加社会实践，将书本知识与实践知识相结合，使自己有适应社会的能力，应变能力有进一步提高。 我开朗活泼，品行端正，为人正直。能够自觉抵制各种不良习气。我做事积极主动，从不计较个人得失。对学校和班级交给的任务都能认真及时完成。我总是以学校和班级荣誉为重。高中二年级，我又通过公开竞选成为为学校学生会副主席，虽然学生
会事务繁多，但我能够合理安排课堂文化课学习和学生会工作，做到两不误，两头都干好。高二学期末，我光荣的被评为优秀学生会干部，青州二中优秀团员。 我学习态度端正，能够掌握正确的学习方法。课堂上认真听讲，目的明确，做好笔记，注重理解和掌握，强化练习，学会分类归纳，不断总结，摸索出适合自己的学习方法，养成良好的学习习惯。在学习中知难而进，敢于正视自己，积极主动地思考问题。课下高质量的完成各科作业，并能及时与任课教师交流沟通。 在较好地完成课堂学业同时，我积极发展个人爱好特长。参加学校篮球队，积极刻苦训练，不断提高个人篮球技艺水平，主动参加校内外各种篮球赛事，作为篮球队骨干队员，我勇挑重担，在20xx年代表青州市分别参加潍坊市级篮球赛和淄博市高中学校篮球邀请赛，为母校和青州市争光添彩。在课余时间里，我喜欢博览群书，开拓视野，增长知识，不断充实自己。还利用假期参加电脑培训，多次作为学生志愿者参加各种社会公益活动，自身品格修养能力得到提升和锻炼。 我能够团结其他同学，热心帮助有困难同学。我尊敬师长，孝敬父母。我总是力所能及的做好自己分内的事。我有很强的社会公德心。我行为文明，爱护公物。我为人朴实真诚，生活上勤俭节约，独立性较强。热爱集体，尊敬师长，对学校和班级交给的任务都能认真及时完成。踊跃参加各项社会公益活动，主动投
with的用法大全----四级专项训练with结构是许多英语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 一、 with结构的构成 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二部分补足语由形容词、副词、介词短语、动词不定式或分词充当，分词可以是现在分词，也可以是过去分词。With结构构成方式如下： 1. with或without-名词/代词+形容词; 2. with或without-名词/代词+副词; 3. with或without-名词/代词+介词短语; 4. with或without-名词/代词+动词不定式; 5. with或without-名词/代词+分词。 下面分别举例：
1、 She came into the room，with her nose red because of cold.(with+名词+形容词，作伴随状语) 2、 With the meal over ， we all went home.(with+名词+副词，作时间状语) 3、The master was walking up and down with the ruler under his arm。(with+名词+介词短语，作伴随状语。) The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house，with not a man ，woman or child to say he was kind to me.(with+名词+不定式，作伴随状语) He could not finish it without me to help him.(without+代词 +不定式，作条件状语) 5、She fell asleep with the light burning.(with+名词+现在分词，作伴随状语) 6、Without anything left in the cupboard， she went out to get something to eat.(without+代词+过去分词，作为原因状语) 二、with结构的用法 在句子中with结构多数充当状语，表示行为方式，伴随情况、时间、原因或条件(详见上述例句)。
1.I don`t like a ___(rain) day. 2.Please tell Tom ___(call)me back. 3.____ is the weather like? It`s snowy. How. What. Where. Why. 4.____ ia it going? It`s great. How. What. Where. When. 5.Is he____? No, he’s ___ in the river. A. swims,fishing B. swimming, running C. swim,walking D. running, swimming 6.Hello!is that Linda speaking? Yes.______ Who’s that? Who’s it? Who are you? I’m Linda. 7.He often ____(watch)TV in the evening. But now he _____(read)a book. 8.My uncle is a ___(cook). 9.It’s a nice ____. A weather photo salad rice 10.After supper Mary often ___ along the river. Take a walk takes walk taking a walk takes a walk 11.How’s it going? _____. The weather is so cold. Preety good. Great. Terrible. Not bad. 12.Beijing ___ (have)a long history. 13.I am having a great time ____(visit)my grandmother. 14.We feel very ___ after the ___ (relax) vacation. 15.We are ___ vacation in Dalian. A in B on C at Dof 16.What ____ when it’s sunny? Aare you doing B do you do C does you D is he do 17.Tom like to go ___(skate) 18.Kate is very tired, so she needs ____(relax) 19.Would you like to ____ a photofor us? Sure. Agive B make C take Dget 20.Could you help me put up the maps on the wall?_____ A.No problem B. I hope so C. That’s all right D.That’s a good idea. 21.China plan to let Jim ____the Xisha Islands> Visit visits visiting visited 22.Where is Grace? She ____ IN THE yard. Reads read is reading was reading 23.Britain is ___ European country and Singapore is ___ Asian country A,a,a a, an the , an a ,the 1.I don`t like a ___(rain) day. 2.Please tell Tom ___(call)me back. 3.____ is the weather like? It`s snowy. How. What. Where. Why. 4.____ ia it going? It`s great. How. What. Where. When. 5.Is he____? No, he’s ___ in the river. A. swims,fishing B. swimming, running C. swim,walking D. running, swimming 6.Hello!is that Linda speaking? Yes.______ Who’s that? Who’s it? Who are you? I’m Linda. 7.He often ____(watch)TV in the evening. But now he _____(read)a book. 8.My uncle is a ___(cook). 9.It’s a nice ____. A weather photo salad rice 10.After supper Mary often ___ along the river. Take a walk takes walk taking a walk takes a walk 11.How’s it going? _____. The weather is so cold. Preety good. Great. Terrible. Not bad. 12.Beijing ___ (have)a long history. 13.I am having a great time ____(visit)my grandmother. 14.We feel very ___ after the ___ (relax) vacation. 15.We are ___ vacation in Dalian. A in B on C at Dof 16.What ____ when it’s sunny? Aare you doing B do you do C does you D is he do 17.Tom like to go ___(skate) 18.Kate is very tired, so she needs ____(relax) 19.Would you like to ____ a photofor us? Sure. Agive B make C take Dget 20.Could you help me put up the maps on the wall?_____ A.No problem B. I hope so C. That’s all right D.That’s a good idea. 21.China plan to let Jim ____the Xisha Islands> Visit visits visiting visited 22.Where is Grace? She ____ IN THE yard. Reads read is reading was reading 23.Britain is ___ European country and Singapore is ___ Asian country A,a,a a, an the , an a ,the
with用法小结 一、with表拥有某物 Mary married a man with a lot of money . 马莉嫁给了一个有着很多钱的男人。 I often dream of a big house with a nice garden . 我经常梦想有一个带花园的大房子。 The old man lived with a little dog on the lonely island . 这个老人和一条小狗住在荒岛上。 二、with表用某种工具或手段 I cut the apple with a sharp knife . 我用一把锋利的刀削平果。 Tom drew the picture with a pencil . 汤母用铅笔画画。 三、with表人与人之间的协同关系 make friends with sb talk with sb quarrel with sb struggle with sb fight with sb play with sb work with sb cooperate with sb I have been friends with Tom for ten years since we worked with each other, and I have never quarreled with him . 自从我们一起工作以来，我和汤姆已经是十年的朋友了，我们从没有吵过架。 四、with 表原因或理由 John was in bed with high fever . 约翰因发烧卧床。 He jumped up with joy . 他因高兴跳起来。 Father is often excited with wine . 父亲常因白酒变的兴奋。 五、with 表“带来”，或“带有,具有”，在…身上，在…身边之意
高中毕业自我鉴定范文大全 关于高中毕业自我鉴定范文大全 高中毕业自我鉴定范文(一) 紧张有序的高中生活即将与我告别。回想三年里有过多少酸甜苦辣，曾经付出了多少辛勤的汗水，但也得到了相应的汇报。在老师的启发教导下，我在德智体方面全面发展，逐渐从幼稚走向成熟。 在政治上，我有坚定正确的立场，热爱祖国，热爱党，认真学习并拥护党的各项方针政策，积极要求进步，思想觉悟高，爱憎分明，踊跃参加各项社会公益活动，主动投入捐款救灾行列，用微薄的力量，表达自己的爱心，做一个文明市民。 在学习上，我有刻苦钻研的学习精神，学习态度端正，目的明确，专心上课并做好笔记，注重理解和掌握，强化练习，学会分类归纳，不断总结，摸索出适合自己的学习方法，养成良好的学习习惯。在学习中知难而进，敢于正视自己的弱点并及时纠正，同时我也积极参加社会实践，将书本知识与实践知识相结合，使自己有适应社会的能力，应变能力有进一步提高。在课余时间里，我喜欢博览群书，开拓视野，增长知识，不断充实自己。还利用假期参加电脑培训，并取得结业证书，高三年被评为校级三好生。 生活上，我拥有严谨认真的作风，为人朴实真诚，勤俭节约，生活独立性较强。热爱集体，尊敬师长，团结同学，对班级交给的任务都能认真及时完成。 我的兴趣广泛，爱好体育、绘画等，积极参加各类体育竞赛，达到国家规定的体育锻炼标准。 高中三年生活有使我清醒地认识到自己的不足之处，如有时学习时间抓不紧，各科学习时间安排不尽合理。因此，我将加倍努力，不断改正缺点，挖掘潜力，以开拓进取、热情务实的精神面貌去迎接未来的挑战。 高中毕业自我鉴定范文(二) 时间过得太快，在我还没反应过来的时候，丰富多彩的三年高中生活就即将结束了，这三年是我人生中最重要的一段里程，它将永远铭记在我的脑海里。 我衷心拥护中国共产党的领导，热爱蒸蒸日上、迈着改革步伐前进的社会主义祖国，用建设有中国特色的社会主义理论武装自己，积极参加党章学习小组，逐步提高自己的政治思想觉悟，关心时事政治，思想健康进步。自觉遵守《中学生守则》和《中学生日常行为规范》。 在学习上，我有刻苦钻研的学习精神，学习态度端正，目的明确，专心上课并做好笔记，注重理解和掌握，强化练习，学会分类归纳，不断总结，摸索出适合自己的学习方法，养成良好的学习习惯。在学习中知难而进，敢于正视自己的弱点并及时纠正，同时我也积极参加社会实践，将书本知识与实践知识相结合，使自己有适应社会的能力，应变能力有进一步提高。 生活上，我拥有严谨认真的作风，为人朴实真诚，勤俭节约，生活独立性较强。热爱集体，尊敬师长，团结同学，对班级交给的任务都能认真及时完成。我的兴趣广泛，爱好体育、绘画等。 作为跨世纪的一代，我们即将告别中学时代的酸甜苦辣，迈入高校去寻找另一片更加广阔的天空。在这最后的中学生活里，我将努力完善自我，提高学习成绩，为几年来的中学生活划上完美的句号，也以此为人生篇章中光辉的一页。 高中毕业自我鉴定范文(三) 时光如梭，转眼即逝，高考已经结束，眼见着高中就要毕业，当我回首这三年来的生活时，才发现这三年来的生活竟是历历在目：
in，on ，at，by ，of ，with 介词区别与用法 in用在年月季节前，还有上午、下午等固定习语里 at用于传统的节日前，如at Christmas等；还有固定词组：at noon, at night;在点时间前用at 如at 7.15 on 用于具体的日期前，星期几，几号，包括那天的上午下午晚上等，如on Friday afternoon with: 一、with表拥有某物 Mary married a man with a lot of money . 马莉嫁给了一个有着很多钱的男人。 I often dream of a big house with a nice garden . 我经常梦想有一个带花园的大房子。 The old man lived with a little dog on the lonely island . 这个老人和一条小狗住在荒岛上。 二、with表用某种工具或手段 I cut the apple with a sharp knife . 我用一把锋利的刀削平果。 Tom drew the picture with a pencil . 汤母用铅笔画画。 三、with表人与人之间的协同关系 make friends with sb talk with sb quarrel with sb struggle with sb fight with sb play with sb work with sb cooperate with sb I have been friends with Tom for ten years since we worked with each other , and I have never quarreled with him . 自从我们一起工作以来，我和汤母已经是十年的朋友了，但我们从没有吵过架。 四、with 表原因或理由 John was in bed with high fever . 约翰因发烧卧床。 He jumped up with joy . 他因高兴跳起来。 Father is often excited with wine . 父亲常因白酒变的兴奋。 五、with 表“带来”，或“带有,具有”，在…身上，在…身边之意 The girl with golden hair looks beautiful . 那个金头发的女孩看起来漂亮。 The famous director will come to the meeting with the leading actor and actress .
高考个人自我鉴定范文 第一篇：高考个人自我鉴定内容 紧张有序的高中生活即将与我告别。回想三年里有过多少酸甜苦辣，曾经付出了多少辛勤的汗水，但也得到了相应的汇报。 在老师的启发教导下，我在德智体方面全面发展，逐渐从幼稚走向成熟。 在政治上，我有坚定正确的立场，热爱祖国，热爱党，认真学习并拥护党的各项方针政策，积极要求进步，思想觉悟高，爱憎分明，踊跃参加各项社会公益活动，主动投入捐款救灾行列，用微薄的力量，表达自己的爱心，做一个文明市民。 在学习上，我有刻苦钻研的学习精神，学习态度端正，目的明确，专心上课并做好笔记，注重理解和掌握，强化练习，学会分类归纳，不断总结，摸索出适合自己的学习方法，养成良好的学习习惯。在学习中知难而进，敢于正视自己的弱点并及时纠正，同时我也积极参加社会实践，将书本知识与实践知识相结合，使自己有适应社会的能力，应变能力有进一步提高。在课余时间里，我喜欢博览群书，开拓视野，增长知识，不断充实自己。还利用假期参加电脑培训，并取得结业证书，高三年被评为校级三好生。 生活上，我拥有严谨认真的作风，为人朴实真诚，勤俭节约，生活独立性较强。热爱集体，尊敬师长，团结同学，对班级交给的任务都能认真及时完成。 我的兴趣广泛，爱好体育、绘画等，积极参加各类体育竞赛，达到国家规定的体育锻炼标准。 高中三年生活有使我清醒地认识到自己的不足之处，如有时学习时间抓不紧，各科学习时间安排不尽合理。因此，我将加倍努力，不断改正缺点，挖掘潜力，以开拓进取、热情务实的精神面貌去迎接未来的挑战。 第二篇：高考自我鉴定 高考自我鉴定 时光如梭，转眼即逝，高考自我鉴定。当毕业在即，回首三年学习生活，历历在目、、、
介词with的用法大全 With是个介词，基本的意思是“用”，但它也可以协助构成一个极为多采多姿的句型，在句子中起两种作用；副词与形容词。 with在下列结构中起副词作用： 1.“with+宾语+现在分词或短语”，如： (1) This article deals with common social ills, with particular attention being paid to vandalism. 2.“with+宾语+过去分词或短语”，如： (2) With different techniques used, different results can be obtained. (3) The TV mechanic entered the factory with tools carried in both hands. 3.“with+宾语+形容词或短语”，如： (4) With so much water vapour present in the room, some iron-made utensils have become rusty easily. (5) Every night, Helen sleeps with all the windows open. 4.“with+宾语＋介词短语”，如： (6) With the school badge on his shirt, he looks all the more serious. (7) With the security guard near the gate no bad character could do any thing illegal. 5.“with+宾语＋副词虚词”，如： (8) You cannot leave the machine there with electric power on. (9) How can you lock the door with your guests in? 上面五种“with”结构的副词功能，相当普遍，尤其是在科技英语中。 接着谈“with”结构的形容词功能，有下列五种： 一、“with+宾语+现在分词或短语”，如： (10) The body with a constant force acting on it. moves at constant pace. (11) Can you see the huge box with a long handle attaching to it ? 二、“with+宾语+过去分词或短语” (12) Throw away the container with its cover sealed. (13) Atoms with the outer layer filled with electrons do not form compounds. 三、“with+宾语+形容词或短语”，如： (14) Put the documents in the filing container with all the drawers open.
With复合结构的用法小结 with结构是许多英语复合结构中最常用的一种。学好它对学好复合宾语结构、不定式复合结构、动名词复合结构和独立主格结构均能起很重要的作用。本文就此的构成、特点及用法等作一较全面阐述，以帮助同学们掌握这一重要的语法知识。 一、with结构的构成 它是由介词with或without+复合结构构成，复合结构作介词with或without的复合宾语，复合宾语中第一部分宾语由名词或代词充当，第二 部分补足语由形容词、副词、介词短语、动词不定式或分词充当，分词可以是现在分词，也可以是过去分词。With结构构成方式如下： 1. with或without-名词/代词+形容词； 2. with或without-名词/代词+副词； 3. with或without-名词/代词+介词短语； 4. with或without-名词/代词+动词不定式； 5. with或without-名词/代词+分词。 下面分别举例： 1、She came into the room，with her nose red because of cold.（with+名词+形容词，作伴随状语） 2、With the meal over ，we all went home.（with+名词+副词，作时间状语） 3、The master was walking up and down with the ruler under his arm。（with+名词+介词短语，作伴随状语。）The teacher entered the classroom with a book in his hand. 4、He lay in the dark empty house，with not a man ，woman or child to say he was kind to me.（with+名词+不定式，作伴随状语）He could not finish it without me to help him.（without+代词+不定式，作条件状语） 5、She fell asleep with the light burning.（with+名词+现在分词，作伴随状语）Without anything left in the cupboard，shewent out to get something to eat.（without+代词+过去分词，作为原因状语） 二、with结构的用法 在句子中with结构多数充当状语，表示行为方式，伴随情况、时间、原因或条件（详见上述例句）。 With结构在句中也可以作定语。例如： 1.I like eating the mooncakes with eggs. 2.From space the earth looks like a huge water-covered globe with a few patches of land sticking out above the water. 3.A little boy with two of his front teeth missing ran into the house. 三、with结构的特点 1. with结构由介词with或without+复合结构构成。复合结构中第一部分与第二部分语法上是宾语和宾语补足语关系，而在逻辑上，却具有主谓关系，也就是说，可以用第一部分作主语，第二部分作谓语，构成一个句子。例如：With him taken care of，we felt quite relieved.（欣慰）→（He was taken good care of.）She fell asleep with the light burning. →（The light was burning.）With her hair gone，there could be no use for them. →（Her hair was gone.） 2. 在with结构中，第一部分为人称代词时，则该用宾格代词。例如：He could not finish it without me to help him. 四、几点说明： 1. with结构在句子中的位置：with 结构在句中作状语，表示时间、条件、原因时一般放在
高中毕业生自我鉴定样板（一） 三年来，学习上我严格要求自己，注意摸索适合自己情况的学习方法，积极思维，分析、解决问题能力强，学习成绩优良。我遵纪守法，尊敬师长，热心助人，与同学相处融洽。我有较强的集体荣誉感，努力为班为校做好事。作为一名团员，我思想进步，遵守社会公德，积极投身实践，关心国家大事。在团组织的领导下，力求更好地锻炼自己，提高自己的思想觉悟。 我的性格活泼开朗，积极参加各种有益活动。高一年担任语文科代表，协助老师做好各项工作。参加市演讲比赛获三等奖。高二以来任班级文娱委员，组织同学参加各种活动，如：课间歌咏，班级联欢会，集体舞赛等。在校文艺汇演中任领唱，参加朗诵、小提琴表演。在校辩论赛在表现较出色，获“最佳辩手”称号。我爱好运动，积极参加体育锻炼，力求德、智、体全面发展，校运会上，在800米、200米及4×100米接力赛中均获较好名次。 三年的高中生活，使我增长了知识，也培养了我各方面的能力，为日后我成为社会有用的人打下了坚实的基础。通过三年的学习，我也发现了自己更大的潜能，具体体现在我学会了合理安排时间，学会了更有序有效的处理生活、学习及人际交往的关系。在我发现自己的潜能后，我尽力锻炼身体，磨砺意志，培养吃苦精神，完善自我。从而保证日后的学习成绩能有较大幅度的提高。 我们即将告别中学时代的酸甜苦辣，迈入高校去寻找另一片更加广阔的天空。在这最后的中学生活里，我将努力完善自我，提高学习成绩，为几年来的中学生活划上完美的句号，也以此为人生篇章中光辉的一页。 高中毕业生自我鉴定（二） 珍贵的三年的高中生活已接近尾声，感觉非常有必要总结一下高中三年的得失，从中继承做得好的方面改进不足的地方，使自己回顾走过的路，也更是为了看清将来要走的路。 学习成绩不是非常好，但我却在学习的过程中收获了很多。首先是我端正了学习态度。在我考进高中时，脑子里想的是好好放松从重压下解放出来的自己，不想考上好的大学，然而很快我就明白了，高中需努力认真的学习。在老师的谆谆教导下，看到周围的同学们拼命的学习，我也打消了初衷，开始高中的学习旅程。懂得了运用学习方法同时注重独立思考。要想学好只埋头苦学是不行的，要学会“方法”，做事情的方法。古话说的好，授人以鱼不如授人以渔，我来这里的目的就是要学会“渔”，但说起来容易做起来难，我换了好多种方法，做什么都勤于思考，遇有不懂的地方能勤于请教。在学习时，以“独立思考”作为自己的座右铭，时刻不忘警戒。随着学习的进步，我不止是学到了课本知识，我的心智也有了一个质的飞跃，我认为这对于将来很重要。在学习知识这段时间里，我更与老师建立了浓厚的师生情谊。老师们的谆谆教导，使我体会了学习的乐趣。我与身边许多同学，也建立了良好的学习关系，互帮互助，克服难关。我在三年的高中学习中，我认真积极参加每次实验，锻炼了自我的动手和分析问题能力，受益匪浅。 一直在追求人格的升华，注重自己的品行。我崇拜有巨大人格魅力的人，并一直希望自己也能做到。在三年中，我坚持着自我反省且努力的完善自己的人格。三年中，我读了一些名著和几本完善人格的书，对自己有所帮助，越来越认识到品行对一个人来说是多么的重要，关系到是否能形成正确的人生观世界观。所以无论在什么情况下，我都以品德至上来要求自己。无论何时何地我都奉行严于律己的信条，并切实的遵行它。平时友爱同学，尊师重道，乐于助人。以前只是觉得帮助别人感到很开心，是一种传统美德。现在我理解道理，乐于助人不仅能铸造高尚的品德，而且自身也会得到很多利益，帮助别人的同时也是在帮助自己。回顾三年高中生活，我很高兴能在同学有困难的时候曾经帮助过他们，相对的，在我有困难
介词“with”的用法 1、同, 与, 和, 跟 talk with a friend 与朋友谈话 learn farming with an old peasant 跟老农学习种田 fight [quarrel, argue] with sb. 跟某人打架 [争吵, 辩论] [说明表示动作的词, 表示伴随]随着, 和...同时 change with the temperature 随着温度而变化 increase with years 逐年增加 be up with the dawn 黎明即起 W-these words he left the room. 他说完这些话便离开了房间。2 2、表示使用的工具, 手段 defend the motherland with one s life 用生命保卫祖国 dig with a pick 用镐挖掘 cut meat with a knife 用刀割肉3
3、说明名词, 表示事物的附属部分或所具有的性质]具有; 带有; 加上; 包括...在内 tea with sugar 加糖的茶水 a country with a long history 历史悠久的国家4 4、表示一致]在...一边, 与...一致; 拥护, 有利于 vote with sb. 投票赞成某人 with的复合结构作独立主格,表示伴随情况时，既可用分词的独立结构，也可用with的复合结构： with +名词(代词)+现在分词/过去分词/形容词/副词/不定式/介词短语。例如： He stood there, his hand raised. = He stood there, with his hand raise.他举手着站在那儿。 典型例题 The murderer was brought in, with his hands ___ behind his back A. being tied B. having tied C. to be tied D. tied 答案D. with +名词(代词)+分词+介词短语结构。当分词表示伴随状况时，其主语常常用