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# 经济代写|博弈论代考Game theory代写|Dominance and Best Response Compared

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## 经济代写|博弈论代考Game theory代写|Dominance and Best Response Compared

There is a precise relation between dominance and best response, the latter of which underpins the theories of behavior to come. For a given game, let $U D_i$ be the set of strategies for player $i$ that are not strictly dominated. Let $B_i$ be the set of strategies for player $i$ that are best responses, over all of the possible beliefs of player $i$. Mathematically,
$$B_i=\left{s_i \mid \text { there is a belief } \theta_{-i} \in \Delta S_{-i} \text { such that } s_i \in B R_i\left(\theta_{-i}\right)\right} .$$
That is, if a strategy $s_i$ is a best response to some possible belief of player $i$, then $s_i$ is contained in $B_i$. As heretofore noted, the notion of best response will be of primary importance. Unfortunately, determining the set $B_i$ is sometimes a greater chore than determining $U D_i$. Fortunately, the two sets are closely related.
To build your intuition, examine the game in Figure 6.4. Note first that $R$ is dominated for player 2.9 ${ }^9$ Thus, $U D_2={L}$ in this game. Also note that strategy $\mathrm{R}$ can never be a best response for player 2, because $\mathrm{L}$ yields a strictly higher payoff regardless of what player 1 does. In other words, for any belief of player 2 about player 1’s behavior, player 2 ‘s only best response is to select $\mathrm{L}$. Therefore $B_2={L}$. Obviously, $B_2=U D_2$ and, for this game, dominance and best response yield the same conclusion about rational behavior for player 2 .

## 经济代写|博弈论代考Game theory代写|Weak dominance

Recall that a key aspect of the dominance definition is the strict inequality, so that a mixed strategy $\sigma_i$ is said to dominate a pure strategy $s_i$ if and only if $u_i\left(\sigma_i, s_{-i}\right)>u_i\left(s_i, s_{-i}\right)$ for all $s_{-i} \in S_{-i}$. One can also consider a version of dominance based on a weaker condition: We say that mixed strategy $\sigma_i$ weakly dominates pure strategy $s_i$ if $u_i\left(\sigma_i, s_{-i}\right) \geq u_i\left(s_i, s_{-i}\right)$ for all $s_{-i} \in S_{-i}$ and $u_i\left(\sigma_i, s_{-i}^{\prime}\right)>u_i\left(s_i, s_{-i}^{\prime}\right)$ for at least one strategy $s_{-i}^{\prime} \in S_{-i}$ of the other players. In other words, $\sigma_i$ performs at least as well as does strategy $s_i$, and it performs strictly better against at least one way in which the other players may play the game.
Figure 6.6 provides an illustration of weak dominance. In the game pictured, if player 1 were to select $Y$, then player 2’s strategy M delivers a strictly higher payoff than does $\mathrm{L}$. If player 1 selects $\mathrm{X}$, then strategies $\mathrm{L}$ and $\mathrm{M}$ yield the same payoff for player 2 . Thus, player 2 always weakly prefers $\mathrm{M}$ to $\mathrm{L}$, and she strictly prefers $\mathrm{M}$ in the event that player 1 picks $\mathrm{Y}$. This means that $\mathrm{M}$ weakly dominates $\mathrm{L}$.

In relation to best-response behavior, weak dominance embodies a form of cautiousness, as though the players cannot be too sure about each other’s strategies. In the example of Figure 6.6, player 2 might reasonably select $\mathrm{L}$ if she is certain that player 1 will choose $X$. On the other hand, if she entertains the slightest doubt-putting any small, strictly positive probability on $\mathrm{Y}$-then $\mathrm{M}$ is her only best response. The example suggests a general relation between weak dominance and best responses to “cautious” beliefs. To make this formal, for any game let $W U D_i$ be the set of strategies for player $i$ that are not weakly dominated. Call a belief $\theta_{-i}$ fully mixed if $\theta_{-i}\left(s_{-i}\right)>0$ for all $s_{-i} \in S_{-i}$. This simply means that $\theta_{-i}$ puts positive probability on every strategy profile of the other players. Then let $B_i^{c f}$ be the set of strategies for player $i$ that are best responses to fully mixed beliefs. In the superscript, $c$ denotes that correlated conjectures are allowed, and $f$ denotes that beliefs are fully mixed.
Result: For any finite game, $B_i^{c f}=W U D_i$ for each player $i=1,2, \ldots, n$.

# 博弈论代写

## 经济代写|博弈论代考Game theory代写|Dominance and Best Response Compared

$$B_i=\left{s_i \mid \text { there is a belief } \theta_{-i} \in \Delta S_{-i} \text { such that } s_i \in B R_i\left(\theta_{-i}\right)\right} .$$

## 经济代写|博弈论代考Game theory代写|Weak dominance

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## MATLAB代写

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