User:IssaRice/Minus notation in game theory - Revision history

IssaRice at 23:07, 13 August 2018

2018-08-13T23:07:54Z

← Older revision		Revision as of 23:07, 13 August 2018
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	* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)		* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)
	* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math> (if the <math>i</math>th strategy needs to be put in the correct place) and <math>v_i(s)</math> (is the strategies are already in order). Note that the altered <math>v_i</math> suggested in the previous bullet is, in this notation, <math>v_i \circ g_i</math>.		* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math> (if the <math>i</math>th strategy needs to be put in the correct place) and <math>v_i(s)</math> (if the strategies are already in order). Note that the altered <math>v_i</math> suggested in the previous bullet is, in this notation, <math>v_i \circ g_i</math>.

	==References==		==References==

	<references/>		<references/>

IssaRice at 23:07, 13 August 2018

2018-08-13T23:07:07Z

← Older revision		Revision as of 23:07, 13 August 2018
Line 10:		Line 10:

	* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)		* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)
	* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math>. Note that the altered <math>v_i</math> suggested in the previous bullet is, in this notation, <math>v_i \circ g_i</math>.		* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math> (if the <math>i</math>th strategy needs to be put in the correct place) and <math>v_i(s)</math> (is the strategies are already in order). Note that the altered <math>v_i</math> suggested in the previous bullet is, in this notation, <math>v_i \circ g_i</math>.

	==References==		==References==

	<references/>		<references/>

IssaRice at 23:04, 13 August 2018

2018-08-13T23:04:58Z

← Older revision		Revision as of 23:04, 13 August 2018
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	* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)		* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)
	* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math>.		* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math>. Note that the altered <math>v_i</math> suggested in the previous bullet is, in this notation, <math>v_i \circ g_i</math>.

	==References==		==References==

	<references/>		<references/>

IssaRice at 23:02, 13 August 2018

2018-08-13T23:02:22Z

← Older revision		Revision as of 23:02, 13 August 2018
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	But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!		But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!

	I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),<ref>To be even more precise, given finite sets of strategies <math>S_1,\ldots,S_n</math>, one can start with <math>S_1</math> and replace each strategy <math>s_i</math> with <math>(i, s_i)</math>, to number them. Then for <math>S_2</math>, one starts the numbering where <math>S_1</math> left off, and so on.</ref> so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.		I think usually this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),<ref>To be even more precise, given finite sets of strategies <math>S_1,\ldots,S_n</math>, one can start with <math>S_1</math> and replace each strategy <math>s_i</math> with <math>(i, s_i)</math>, to number them. Then for <math>S_2</math>, one starts the numbering where <math>S_1</math> left off, and so on.</ref> so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.

	Here are two more ideas for dealing with this notation:		Here are two more ideas for dealing with this notation:

IssaRice at 23:02, 13 August 2018

2018-08-13T23:02:03Z

← Older revision		Revision as of 23:02, 13 August 2018
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	I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),<ref>To be even more precise, given finite sets of strategies <math>S_1,\ldots,S_n</math>, one can start with <math>S_1</math> and replace each strategy <math>s_i</math> with <math>(i, s_i)</math>, to number them. Then for <math>S_2</math>, one starts the numbering where <math>S_1</math> left off, and so on.</ref> so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.		I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),<ref>To be even more precise, given finite sets of strategies <math>S_1,\ldots,S_n</math>, one can start with <math>S_1</math> and replace each strategy <math>s_i</math> with <math>(i, s_i)</math>, to number them. Then for <math>S_2</math>, one starts the numbering where <math>S_1</math> left off, and so on.</ref> so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.

			Here are two more ideas for dealing with this notation:

			* One can change each <math>v_i</math> to have domain <math>S_i\times S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. This is sort of inelegant because now one cannot pass <math>s \in S_1\times \cdots \times S_n</math> to an arbitrary <math>v_i</math>. (One can only pass it to <math>v_1</math>.)
			* One can define a function <math>g_i</math> for each player that is supposed to "put <math>s_i</math> back in the right spot". Formally, <math>g_i\colon S_i \times (S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n) \to S_1 \times \cdots \times S_n</math> is defined by <math>g(x,y) = (y_{\leq i-1}, x, y_{\geq i+1})</math>. Now one can write <math>v_i(g_i(s_i, s_{-i}))</math>.

			==References==

	<references/>		<references/>

IssaRice at 22:50, 13 August 2018

2018-08-13T22:50:18Z

← Older revision		Revision as of 22:50, 13 August 2018
Line 5:		Line 5:
	But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!		But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!

	I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable), so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.		I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable),<ref>To be even more precise, given finite sets of strategies <math>S_1,\ldots,S_n</math>, one can start with <math>S_1</math> and replace each strategy <math>s_i</math> with <math>(i, s_i)</math>, to number them. Then for <math>S_2</math>, one starts the numbering where <math>S_1</math> left off, and so on.</ref> so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.

			<references/>

IssaRice at 22:46, 13 August 2018

2018-08-13T22:46:31Z

← Older revision		Revision as of 22:46, 13 August 2018
Line 5:		Line 5:
	But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!		But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!

	I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable), so that one can pass a ''set'' of strategies like <math>\{s_1,ldots,s_n\}</math> and there would be no confusion even if the ordering is different.		I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable), so that one can pass a ''set'' of strategies like <math>\{s_1, \ldots ,s_n\}</math> and there would be no confusion even if the ordering is different.

IssaRice: Created page with "In game theory, $s_{-i}$ is a shorthand for $(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S..."$

2018-08-13T22:45:36Z

Created page with "In game theory, <math>s_{-i}</math> is a shorthand for <math>(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S..."

New page

In game theory, <math>s_{-i}</math> is a shorthand for <math>(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S_n</math>. In other words, <math>s_{-i}</math> is a tuple of strategies of all the players other than player <math>i</math>.

One then writes things like <math>v_i(s_i, s_{-i})</math> for the payoff of the strategy profile where each player <math>i</math> chooses <math>s_i</math>.

But to my mind, there is a type error here, because <math>v_i \colon S_1 \times \cdots \times S_n \to \mathbf R</math> is supposed to take the strategies in order, whereas in the previous paragraph, for each player, their own strategy appears first. To give an example, in the case of <math>n=2</math>, each <math>v_i</math> should take <math>(s_1,s_2) \in S_1\times S_2</math>, but the notation <math>(s_i, s_{-i})</math> for player 2 is <math>(s_2,s_1)</math>, which is reversed. Type error!

I think usually, this is not a problem, because the reader can mentally put the <math>s_i</math> in the right spot, or one can assume that no two strategies (even across different players) are the same (e.g. the strategies can be "marked" with numbers to be distinguishable), so that one can pass a ''set'' of strategies like <math>\{s_1,ldots,s_n\}</math> and there would be no confusion even if the ordering is different.

User:IssaRice/Minus notation in game theory - Revision history

IssaRice at 23:07, 13 August 2018

IssaRice at 23:07, 13 August 2018

IssaRice at 23:04, 13 August 2018

IssaRice at 23:02, 13 August 2018

IssaRice at 23:02, 13 August 2018

IssaRice at 22:50, 13 August 2018

IssaRice at 22:46, 13 August 2018

IssaRice: Created page with "In game theory, s_{-i} is a shorthand for (s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S..."

IssaRice: Created page with "In game theory, $s_{-i}$ is a shorthand for $(s_1,\ldots,s_{i-1},s_{i+1},\ldots,s_n) \in S_1 \times \cdots \times S_{i-1} \times S_{i+1} \times \cdots \times S..."$