经济代写|博弈论代写Game Theory代考|ECON90022

经济代写|博弈论代写Game Theory代考|Interaction, Negotiation, and Learning

Most of the two-player games in Chapter 3 have simultaneous moves, where both players choose their actions without knowing the action of the partner and do not change their choice once the action of the partner becomes known. These are referred to as one-shot, simultaneous, or sealed-bid games. We have already argued that real interactions are often not of this form (e.g. Section 3.5). Instead there is a sequence of decisions where each decision is contingent on the current information, and in particular on what has gone before. In this interaction over time there is often learning about the partner or the situation.

For example, consider the decision of parents whether to care for their young or to desert. McNamara et al. (2002) review evidence that the decisions of the two parents are often not independent. One of the cases they consider is that of the penduline tit. In the data reproduced in Fig. 8.1, out of $n=130$ clutches the proportion in which the male cares is $p_m=(0+25) / 130=0.19$. Similarly the proportion in which the female cares is $p_f=(0+67) / 130=0.52$. If individuals are following a mixed-strategy Nash equilibrium with independent choices, then from these marginal proportions an estimate of the number of clutches in which there is biparental care is $n_{\text {bipar }}=$ $n p_m p_f=130 \times(25 / 130) \times(67 / 130)=12.9$. A comparison of observed values with those expected under independence provides strong evidence that decisions are not independent (the null hypothesis of independence can be rejected at $p<0.0001$ in a chi-squared test). It is striking that there are no cases of biparental care. The obvious inference is that if one parent is caring the other notices this and leaves.

This chapter is concerned with the interaction process. By process we mean the sequence of actions and the information on which each action is based. As we show, one cannot predict the outcome of an interaction from the payoffs alone, but one must know the process. For example, just changing the order in which players choose their actions can completely change predictions (Section 8.2). Changing the order of choice has the effect of changing the information available to players. An individual can gain an advantage by giving reliable information to a partner. This might be achieved by individuals committing themselves to a course of action by deliberately handicapping themselves (Section 8.3), or through reputation effects. In an interaction individuals may differ in their abilities, energy reserves, or other aspects of state. Often the state of an individual is private information to that individual that is partly revealed by its actions. The interaction may then be a process in which the individuals gain partial information about one another, although it may be in the interests of each to hide information. So for example, it can be advantageous to a parent to appear less able to care than is actually the case, to force the mate to increase its care effort in order to compensate (Sections $8.4$ and 8.5). It may even be the case that individuals do not know their own state and must learn about that, for instance when establishing dominance relations in a social group. This and other learning scenarios are considered in Sections 8.5, 8.6, and 8.7. Learning mechanisms can be especially helpful to study interactions over time when individuals respond to each other’s actions that in turn are influenced by individual characteristics. Such a large-worlds perspective (Chapter 5) could also be realistic in terms of the traits and mechanisms assumed. We end the chapter (Section 8.8) with a general discussion of game theory for interaction over time, including the role of behavioural mechanisms and how one can think about evolutionary stability for large-worlds models.

经济代写|博弈论代写Game Theory代考|Information and the Order of Choice

Consider two parents that have produced young together. Suppose that each can either care for the young (C) or desert (D). The advantage of caring is that the young do better, which benefits both parents. However, care is costly and reduces prospects for remating. For illustrative purposes, let the payoffs to the two parents be given by Table 8.1. This payoff structure might be appropriate if the male pays a greater cost for uniparental care than the female and also gains less from desertion, perhaps because the local population is male biased. We compare two processes by which the actions are chosen. Suppose first that each parent chooses its action without knowledge of the decision of the other parent. Each then sticks to its decision once the decision of the other becomes known. In game theory this scenario is known as simultaneous moves. From Table $8.1$ it can be seen that whatever the action of the female the male does best if he cares. Given that the male will care the female’s best action is to desert. Thus the only Nash equilibrium is for the male to always care and the female to always desert.
Now suppose that the male chooses his action first. The female then chooses her action knowing the choice of the male. A game with this information structure, where one player makes its decision knowing that of the partner, is known as a Stackelberg game. We can represent this scenario by the game tree in Fig. 8.2. Note that the payoff structure is exactly as in Table 8.1. We can find a Nash equilibrium by working backwards. If the male cares the female will desert and the male will get payoff 3 . If the male deserts the female will care and the male will get payoff 4 . Thus the male should desert and the female will then care. The Nash equilibrium found by working backwards in this way is called a subgame perfect equilibrium (see Section 8.3).
If we compare the outcomes of the two processes we see they predict opposite forms of care. In the simultaneous choice case the male cares, in the sequential choice case he deserts, despite the fact that his best action is to care for either of the fixed actions of the female. This example shows that, by itself, the payoff structure is not enough to predict evolutionary outcomes; we must also specify the decision process. The evidence in Fig. $8.1$ suggests that the penduline tits do not make simultaneous decisions. The process is not, however, liable to be as simple as one of the parents choosing first and the other reacting to this choice (Griggio et al., 2004; van Dijk et al., 2007). For example, during foraging trips each may assess the possibilities to remate if they desert, so that actions may also be contingent on this information. Furthermore, there is evidence that the female attempts to hide the fact she has already produced eggs from the male (Valera et al., 1997). This may allow her to gain an advantage by being the first to choose.

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经济代写|博弈论代写博弈论代考|信息和选择的顺序

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