︠bd3b9eea-1eec-4328-9535-f0ba44e5c2bci︠
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<p>Code that defines a very simple function entitled "TwobyTwoNashEquilibria" and uses simple algebra to compute the equilibria for a 2 by 2 bi-matrix game.</p>

︡44fd23b2-766b-4bca-a5bf-5fd88a7a170f︡{"html": "<p>Code that defines a very simple function entitled \"TwobyTwoNashEquilibria\" and uses simple algebra to compute the equilibria for a 2 by 2 bi-matrix game.</p>"}︡
︠231d03db-788b-4ea9-a9ba-0cd5bbcfce46︠
def TwobyTwoNashEquilibria(A,B=0):
        "Computes the Equilibria for a general 2 by 2 bi matrix game defined by $(A,B)\in\mathbb{R}^{2\\times2}\\times\mathbb{R}^{2\\times 2}$. If B is omitted from input, a zero sum game is assumed."
	A=matrix(A)
	if B==0:
		B=-A
	else:
		B=matrix(B)
	if max(A[0]-A[1])<0: # This implies that the second row strategy dominates the first row strategy
		p=0
		if B[1,0]<=B[1,1]:
			q=0
		else:
			q=1
	elif min(A[0]-A[1])>0: # This implies that the first row strategy dominates the second row strategy
		p=1
		if B[0,0]<=B[0,1]:
			q=0
		else:
			q=1
	elif max(transpose(B)[0]-transpose(B)[1])<0: # This implies that the second column strategy dominates the first column strategy
		q=0
		if A[0,1]<=A[1,1]:
			p=0
		else:
			p=1
	elif min(transpose(B)[0]-transpose(B)[1])>0: # This implies that the first column strategy dominates the second column strategy
		q=0
		if A[0,0]<=A[1,0]:
			p=0
		else:
			p=1
	else: #This computes 0<p<1 and 0<q<1 that ensure equilibria in the case of no dominated strategies. 
		p=(B[1,1]-B[1,0])/(B[0,0]+B[1,1]-B[0,1]-B[1,0])
		q=(A[1,1]-A[0,1])/(A[0,0]+A[1,1]-A[0,1]-A[1,0])
	return [(p,1-p),(q,1-q)]

TwobyTwoNashEquilibria?
︡a202b58f-8e14-467a-8b19-5ad55914a45b︡{"html": "<!--notruncate-->\n\n<div class=\"docstring\">\n    \n  <p><strong>File:</strong> /tmp/tmpXNMwn5/___code___.py</p>\n<p><strong>Type:</strong> &lt;type &#8216;function&#8217;&gt;</p>\n<p><strong>Definition:</strong> TwobyTwoNashEquilibria(A, B=0)</p>\n<p><strong>Docstring:</strong></p>\n<p>Computes the Equilibria for a general 2 by 2 bi matrix game defined by <span class=\"math\">(A,B)\\in\\mathbb{R}^{2\\times2}\\times\\mathbb{R}^{2\\times 2}</span>. If B is omitted from input, a zero sum game is assumed.</p>\n\n\n</div>\n"}︡
︠5c79813b-7624-47c4-bef0-a2e641638690i︠
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<p>Obtaining the equilibria strategies for:&nbsp;<a href="http://en.wikipedia.org/wiki/Matching_pennies" target="_blank">matching pennies</a></p>

︡26d2406a-4d79-461b-8fb3-ef352f46ad1c︡{"html": "<p>Obtaining the equilibria strategies for:&nbsp;<a href=\"http://en.wikipedia.org/wiki/Matching_pennies\" target=\"_blank\">matching pennies</a></p>"}︡
︠7f9c7bc4-8641-43b6-9c50-beebf12376d7︠
TwobyTwoNashEquilibria([[1,-1],[-1,1]])
︡7e6cc32d-9686-4b60-9cd8-6ae3dfb86bff︡{"stdout": "[(1/2, 1/2), (1/2, 1/2)]"}︡
︠db3ff3d3-70a2-48b6-9f69-63bf0bb62f2fi︠
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<p>Obtaining the equilibria strategies for: <a href="http://en.wikipedia.org/wiki/Prisoner's_dilemma" target="_blank">prisoner's dilemma</a></p>

︡2e98e5c8-ac80-4a65-93e5-685631cc65c6︡{"html": "<p>Obtaining the equilibria strategies for: <a href=\"http://en.wikipedia.org/wiki/Prisoner's_dilemma\" target=\"_blank\">prisoner's dilemma</a></p>"}︡
︠51f25728-58d3-4d81-b21b-6d5704ae0dc4︠
TwobyTwoNashEquilibria([[4,1],[5,2]],[[4,5],[1,2]])
︡d10ca6ca-f751-4c6d-ae2b-4b424018ca1c︡{"stdout": "[(0, 1), (0, 1)]"}︡
︠9a69a8e8-5276-42c2-886d-eedac1b32af8i︠
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<p>Obtaining the equilibria strategies for the box kick game of <a href="http://drvinceknight.blogspot.com/2011/12/to-box-kick-or-not-to-box-kick.html" target="_blank">this blog post</a>.</p>

︡959904a2-2c74-45b9-b25b-ad88c31fbc51︡{"html": "<p>Obtaining the equilibria strategies for the box kick game of <a href=\"http://drvinceknight.blogspot.com/2011/12/to-box-kick-or-not-to-box-kick.html\" target=\"_blank\">this blog post</a>.</p>"}︡
︠b579b58e-7802-4377-9fb8-d30b012286ec︠
TwobyTwoNashEquilibria([[20,80],[95,70]])
︡0290abc5-64e4-414b-af55-a0430cf4d146︡{"stdout": "[(5/17, 12/17), (2/17, 15/17)]"}︡