Mixed strategy game - Revision history

Dirkjan at 20:01, 27 August 2020

2020-08-27T20:01:05Z

Dirkjan: Replaced content with "Mixed strategy game Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

2020-08-27T19:20:21Z

Replaced content with "Mixed strategy game Category:mbi Category:Decision Category:M7 Category:Methods Category:Quantitative Category:Term"

Dirkjan: Created page with "In a mixed strategy game the players will select between different strategies. {| class="wikitable" ! ! !B1 !B2 |- |Game player A |A1 |15 |20 |- | |A2 |30 |10 |- |} Player a w..."

2020-08-25T08:05:10Z

Created page with "In a mixed strategy game the players will select between different strategies. {| class="wikitable" ! ! !B1 !B2 |- |Game player A |A1 |15 |20 |- | |A2 |30 |10 |- |} Player a w..."

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In a mixed strategy game the players will select between different strategies.
{| class="wikitable"
!
!
!B1
!B2
|-
|Game player A
|A1
|15
|20
|-
|
|A2
|30
|10
|-
|}
Player a will select strategy A1 as this is the best of the minimum gains (15). Player B will select strategt B2 as this is the smallest of the maxiumum losses. There is no saddle point in this game.
{| class="wikitable"
!
!
!B1
!B2
!Row minimum
|-
|Game player A
|A1
|15
|20
|15
|-
|
|A2
|30
|10
|10
|-
|
|
|30
|20
|
|-
|}Player 'A' can decide which portion of the time it should play strategy A1 and which portion of the time strategy A2 to maximize pay out. This can be though of as a probability. Let p be the proportion of the games that player A selects A1. (1-p) then will represent the percentage of the time that player A selects strategy A2.
If B selects B1 the expected gain is:
15p + 30 (1-p) = 30 - 15p
If B selects B2 the expected gain is:
20p + 10 (1-p) = 10 +10p
Solve this:
{{{
30 - 15p = 10 + 10p
20 = 25p
p = 0.8
}}}
The expected gain for A (assuming B1) is:
{{{
15 * 0.8 + 30 * 0.2 = 18
}}}
And assuming B2:
{{{
20 * 0.8 + 10 * 0.2 = 18
}}}

Same calculations for player B. Let q be the proportion of the time player B selects strategy B1 and (q-1) the proportion of the time he selects B2.
If A selects A1 the expected loss is:
15 * q + 20 (1-q)
If A selects A2 the expected loss is:
30 * q + 10 (1-q)
Solve this:
{{{
15 * q + 20 (1-q) = 30 * q + 10 (1-q)
-5q + 20 = 20q + 10
10 = 25q
q = 0.4
}}}
The expected value for B in this case is also 18.

Part of [[M7-Quantitative Decision Methods-S5 - Game Theory|M7-Quantitative Decision Methods-S5 - Game Theory]]
[[Category:mbi]]

[[Category:Decision]]

[[Category:M7]]

[[Category:Methods]]

[[Category:Quantitative]]

[[Category:Term]]

← Older revision		Revision as of 19:20, 27 August 2020
Line 1:		Line 1:
−	~~In a mixed~~ strategy game ~~the players will select between different strategies.~~	+	Mixed strategy game
−	~~{\| class="wikitable"~~
−	!
−	!
−	~~!B1~~
−	~~!B2~~
−	\|-
−	~~\|Game player A~~
−	~~\|A1~~
−	~~\|15~~
−	~~\|20~~
−	\|-
−	\|
−	~~\|A2~~
−	~~\|30~~
−	~~\|10~~
−	\|-
−	\|}
−	Player a will select strategy A1 as this is the best of the minimum gains (15). Player B will select strategt B2 as this is the smallest of the maxiumum losses. There is no saddle point in this game.
−	~~{\| class="wikitable"~~
−	!
−	!
−	~~!B1~~
−	~~!B2~~
−	~~!Row minimum~~
−	\|-
−	~~\|Game player A~~
−	~~\|A1~~
−	~~\|15~~
−	~~\|20~~
−	~~\|15~~
−	\|-
−	\|
−	~~\|A2~~
−	~~\|30~~
−	~~\|10~~
−	~~\|10~~
−	\|-
−	\|
−	\|
−	~~\|30~~
−	~~\|20~~
−	\|
−	\|-
−	\|}Player 'A' can decide which portion of the time it should play strategy A1 and which portion of the time strategy A2 to maximize pay out. This can be though of as a probability. Let p be the proportion of the games that player A selects A1. (1-p) then will represent the percentage of the time that player A selects strategy A2.
−	~~If B selects B1 the expected gain is:~~
−	~~15p + 30 (1-p) = 30 - 15p~~
−	~~If B selects B2 the expected gain is:~~
−	~~20p + 10 (1-p) = 10 +10p~~
−	~~Solve this:~~
−	~~{{{~~
−	~~30 - 15p = 10 + 10p~~
−	~~20 = 25p~~
−	~~p = 0.8~~
−	~~}}}~~
−	~~The expected gain for A (assuming B1) is:~~
−	~~{{{~~
−	~~15 * 0.8 + 30 * 0.2 = 18~~
−	~~}}}~~
−	~~And assuming B2:~~
−	~~{{{~~
−	~~20 * 0.8 + 10 * 0.2 = 18~~
−	~~}}}~~
−
−	~~Same calculations for player B. Let q be the proportion of the time player B selects strategy B1 and (q-1) the proportion of the time he selects B2.~~
−	~~If A selects A1 the expected loss is:~~
−	~~15 * q + 20 (1-q)~~
−	~~If A selects A2 the expected loss is:~~
−	~~30 * q + 10 (1-q)~~
−	~~Solve this:~~
−	~~{{{~~
−	~~15 * q + 20 (1-q) = 30 * q + 10 (1-q)~~
−	~~-5q + 20 = 20q + 10~~
−	~~10 = 25q~~
−	~~q = 0.4~~
−	~~}}}~~
−	~~The expected value for B in this case is also 18.~~
−
−	~~Part of [[M7-Quantitative Decision Methods-S5 - Game Theory\|M7-Quantitative Decision Methods-S5 - Game Theory]]~~
	[[Category:mbi]]		[[Category:mbi]]

← Older revision		Revision as of 20:01, 27 August 2020
Line 1:		Line 1:
−	~~Mixed~~ strategy game	+	In a mixed strategy game the players will select between different strategies.
		+	{\| class="wikitable"
		+	!
		+	!
		+	!B1
		+	!B2
		+	\|-
		+	\|Game player A
		+	\|A1
		+	\|15
		+	\|20
		+	\|-
		+	\|
		+	\|A2
		+	\|30
		+	\|10
		+	\|-
		+	\|}
		+	Player a will select strategy A1 as this is the best of the minimum gains (15). Player B will select strategt B2 as this is the smallest of the maxiumum losses. There is no saddle point in this game.
		+	{\| class="wikitable"
		+	!
		+	!
		+	!B1
		+	!B2
		+	!Row minimum
		+	\|-
		+	\|Game player A
		+	\|A1
		+	\|15
		+	\|20
		+	\|15
		+	\|-
		+	\|
		+	\|A2
		+	\|30
		+	\|10
		+	\|10
		+	\|-
		+	\|
		+	\|
		+	\|30
		+	\|20
		+	\|
		+	\|-
		+	\|}Player 'A' can decide which portion of the time it should play strategy A1 and which portion of the time strategy A2 to maximize pay out. This can be though of as a probability. Let p be the proportion of the games that player A selects A1. (1-p) then will represent the percentage of the time that player A selects strategy A2.
		+	If B selects B1 the expected gain is:
		+	15p + 30 (1-p) = 30 - 15p
		+	If B selects B2 the expected gain is:
		+	20p + 10 (1-p) = 10 +10p
		+	Solve this:
		+	{{{
		+	30 - 15p = 10 + 10p
		+	20 = 25p
		+	p = 0.8
		+	}}}
		+	The expected gain for A (assuming B1) is:
		+	{{{
		+	15 * 0.8 + 30 * 0.2 = 18
		+	}}}
		+	And assuming B2:
		+	{{{
		+	20 * 0.8 + 10 * 0.2 = 18
		+	}}}
		+
		+	Same calculations for player B. Let q be the proportion of the time player B selects strategy B1 and (q-1) the proportion of the time he selects B2.
		+	If A selects A1 the expected loss is:
		+	15 * q + 20 (1-q)
		+	If A selects A2 the expected loss is:
		+	30 * q + 10 (1-q)
		+	Solve this:
		+	{{{
		+	15 * q + 20 (1-q) = 30 * q + 10 (1-q)
		+	-5q + 20 = 20q + 10
		+	10 = 25q
		+	q = 0.4
		+	}}}
		+	The expected value for B in this case is also 18.
		+
		+	Part of [[M7-Quantitative Decision Methods-S5 - Game Theory\|M7-Quantitative Decision Methods-S5 - Game Theory]]
	[[Category:mbi]]		[[Category:mbi]]