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Famous 2x2 Games: Bi-matrices and Nash Equilibria

Famous 2x2 Games: Bi-matrices and Nash Equilibria
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2015-08-18 12:29
PolygonMatrixOptimizationGeometric Construction
This example demonstrates famous two-player games in which each player has two strategies, represented as 2x2 bi-matrices and analyzed through Nash equilibria. Core concepts include game theory, payoff matrices, strategy profiles, best responses, and Nash equilibrium. In GeoGebra, the construction uses Textfield, inputBox, and TABLE objects to enter players and payoffs, while Segment, PolyLine, and Polygon create the matrix layout or highlighted regions, and If commands test equilibrium conditions. It is suitable for high-school enrichment, introductory university game theory, mathematical modeling, and interactive GeoGebra lessons.

GeoGebra Commands & Parameters

  1. Segment(O, O30) → a
  2. Segment(O01, O31) → b
  3. Segment(O02, O32) → c
  4. Segment(O03, O33) → d
  5. Segment(O, O03) → e
  6. Segment(O10, O13) → f
  7. Segment(O20, O23) → g
  8. Segment(O30, O33) → h
  9. Textfield(Player1) → inputBox1
  10. Textfield(UpperPl) → Player2
  11. Textfield(text3) → inputBox3
  12. Textfield(text4) → inputBox4
  13. Textfield(text5) → Left
  14. Textfield(text6) → Right
  15. Textfield(text7) → inputBox7
  16. Polygon(P, Q, 4) → poly1, i, j, k, l, R, S
  17. Polygon(O, O03, O33, O32, O12, O10) → poly2, o, o03, o33, o32, o12, o10
  18. Polygon(O11, O12, O22, O21) → Cell11, o11, o12_1, o22, o21
  19. Polygon(O21, O22, O32, O31) → Cell12, o21_1, o22_1, o32_1, o31
  20. Polygon(O10, O11, O21, O20) → Cell21, o10_1, o11_1, o21_2, o20
  21. Polygon(O20, O21, O31, O30) → Cell22, o20_1, o21_3, o31_1, o30
  22. Textfield(L11_1) → inputBox8
  23. Textfield(L12_1) → inputBox9
  24. Textfield(L21_1) → inputBox10
  25. Textfield(L22_1) → inputBox11
  26. Textfield(U11_1) → inputBox12
  27. Textfield(U12_1) → inputBox13
  28. Textfield(U21_1) → inputBox14
  29. Textfield(U22_1) → inputBox15
  30. If((L11_1 ≥ L21_1) ∧ (U11_1 ≥ U12_1), 1, 0) → NASH11
  31. If((L22_1 ≥ L12_1) ∧ (U22_1 ≥ U21_1), 1, 0) → NASH22
  32. If((L12_1 ≥ L22_1) ∧ (U12_1 ≥ U11_1), 1, 0) → NASH12
  33. If((L21_1 ≥ L11_1) ∧ (U21_1 ≥ U22_1), 1, 0) → NASH21
  34. Segment(Origin01, Origin11) → n
  35. Segment(Origin11, Origin10) → p
  36. Segment(E_1, F_1) → m
  37. Segment(H_1, I_1) → q
  38. Segment(J_1, K_1) → r
  39. Polygon(Origin, Origin01, Origin11, Origin10) → LeftWhole, origin, origin01, origin11, origin10
  40. Polygon(Origin, Origin01, Origin11, Origin10) → UpperWhole, origin_1, origin01_1, origin11_1, origin10_1
  41. Polygon(Origin, Origin01, Origin11, Origin10) → NashWhole, origin_2, origin01_2, origin11_2, origin10_2
  42. PolyLine(LeftList) → LeftReaction
  43. PolyLine(UpperList) → UpperReaction
  44. PolyLine(NashList) → Nash