ProblemFormulationStability and**Accuracy****Function**sProblem FormulationSensitivity Analysis in Game Theory Yury Nikulin - p. 2/29

Game descriptionProblemFormulationStability and**Accuracy****Function**sWe consider a coalition game with m ≥ 2 players. Let X i be afinite set of (pure) strategies of the playeri ∈ N m := {1, 2, ..., m}. We assume |X i | = 2 for all i ∈ N mindicating each player has a choice of 2 antagonistic strategiesto play. We also assume X i := {0, 1} for all i ∈ N m , i.e. thechoice of each strategy is encoded by means of Booleanvariables x i . The set of all solutions X can be generallydefined as a subset of Cartesian product over all players oftheir sets of strategiesX ⊆ ∏i∈N mX i = {0, 1} m .Observe that now - formally - X is a set of ordered m-tuples.Sensitivity Analysis in Game Theory Yury Nikulin - p. 3/29

Forming a coalition, bansProblemFormulationStability and**Accuracy****Function**sLet s be a number of coalitions in the game and letJ := (J 1 , J 2 , ..., J s ) be a set of coalitions formed. Let alsoX Jr := ∏j∈J rX j , J r ∈ J, r ∈ N s .be a set of strategy combinations accessible by coalition J r .The set of strategy combinations accessible by coalition J r ,with respect to the set of banned strategies B r , is defined as:X BrJ r:= X Jr \B r , J r ∈ J, r ∈ N s .Thus for a m-player s-coalition game with coalition profile(J 1 , J 2 , ..., J s ) and ban profile B := (B 1 , B 2 , ..., B s ), a set ofsolutions is defined as:X := ∏XJ Brr.r∈N sSensitivity Analysis in Game Theory Yury Nikulin - p. 5/29

Feasible solutionsProblemFormulationStability and**Accuracy****Function**sFor any solution x ∗ ∈ X, any coalition J r ∈ J, r ∈ N s , letW Jr (x ∗ ) be a set of all non-banned solutions accessible bycoalition J r from x ∗ by changing strategies of only thoseplayers who are the members of J r :WJ Brr(x ∗ ) := XJ Brr× ∏x ∗ i .i∈N m \J rFor brevity sake, we denote W Jr (x ∗ ) := W BrJ r(x ∗ ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 6/29

EquilibriaProblemFormulationStability and**Accuracy****Function**sWe introduce the notion of equilibrium for the m-players-coalition game with matrix C and coalition profileJ = (J 1 , J 2 , ..., J s ). A solution x ∗ ∈ X is called J-equilibrium if∀ r ∈ N s Λ r (C, J r , x ∗ ) = ∅,where{Λ r (C, J r , x ∗ ) := x ∈ W Jr (x ∗ ) :() ()}∀j ∈ J r p j (C, x) ≤ p j (C, x ∗ ) & ∃j ′ ∈ J r p j ′(C, x) < p j ′(C, x ∗ ) .Sensitivity Analysis in Game Theory Yury Nikulin - p. 7/29

Special casesProblemFormulationStability and**Accuracy****Function**sObviously, if J = ({1}, {2}, ..., {m}) and B r = ∅ for all r ∈ N m(m coalitions, each coalition has one member, no bans), thenthe concept of J-equilibrium transforms into the well-knowconcept of the Nash equilibrium. On the other hand, ifJ = ({1, 2, ..., m}) and B = (∅) (one coalition that contains allplayers, no bans), then the concept of J-equilibrium transformsinto the well-know concept of the Pareto efficiency or Paretoequilibrium.For the game with matrix C, we denote PE m (C, X) andNE m (C, X) the set of Pareto and Nash equilibria, respectively.Sensitivity Analysis in Game Theory Yury Nikulin - p. 8/29

Original problemProblemFormulationStability and**Accuracy****Function**sIt is assumed that the set of strategy profiles X is fixed, i.e wehave coalition and ban profiles J and B fixed, but the originalmatrix of coefficients may vary or be estimated with errors.Moreover, it is assumed that for some originally specifiedmatrix C 0 = {c 0 ij } ∈ Rm×m + we know a J-equilibrium x ∗ whichis an element of the set of all J-equilibria JE(C 0 , X).Sensitivity Analysis in Game Theory Yury Nikulin - p. 9/29

Relative errorProblemFormulationStability and**Accuracy****Function**sNamely, in case of J-equilibrium we introduce forx ∗ ∈ JE(C 0 , X) and a given matrix C ∈ R m×m+ the relativeerror of this solution:ε(C, J, x ∗ ) := maxr∈N smax minx∈W J r (x∗ )p i (C, x ∗ ) − p i (C, x).i∈J r p i (C, x)Observe that for an arbitrary C ∈ R m×m+ we haveε(C, J, x ∗ ) ≥ 0. If ε(C, J, x ∗ ) > 0, then x ∗ ∉ JE(C, X) and thispositive value of the relative error may be treated as a measureof inefficiency of the strategy profile x ∗ for the game with matrixC.The equality ε(C, J, x ∗ ) = 0 formulates in general onlynecessary condition for x ∗ to be J-equilibrium, i.e.ε(C, J, x ∗ ) = 0 does not guarantee that x ∗ ∈ JE m (C, X).Sensitivity Analysis in Game Theory Yury Nikulin - p. 10/29

ExampleProblemFormulationStability and**Accuracy****Function**sExample 1. Let m = 2, s = 1, J = ({1, 2}) and C 0 =(1 22 1Assume that B = ({(0, 0) T , (1, 1) T }), i.e. X = {x 1 , x 2 },x 1 = (0, 1) T , x 2 = (1, 0) T . Then P(C 0 , X) = {(2, 1) T , (1, 2) T }.If we consider the matrixC =(1 12 1),).then P(C, X) = {(1, 1) T , (1, 2) T }. Evidently, x 2 ∈ PE 2 (C 0 , X)and ε(C 0 , J, x 2 ) = 0, but x 2 ∉ PE 2 (C, X) and ε(C, J, x 2 ) = 0.Sensitivity Analysis in Game Theory Yury Nikulin - p. 11/29

ProblemFormulationStability and**Accuracy****Function**sStability and **Accuracy** **Function**sSensitivity Analysis in Game Theory Yury Nikulin - p. 12/29

Stability **Function**ProblemFormulationStability and**Accuracy****Function**sFor a given ρ ∈ [0, q(C 0 , X)), whereq(C 0 , X) := min{c 0 ij , i ∈ N m, j ∈ N m }, we consider a setΩ ρ (C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| ≤ ρ, i ∈ N m , j ∈ N m }.For a J-equilibrium x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), thevalue of the stability function is defined as follows:S(C 0 , J, x ∗ , ρ) :=max ε(C, J,C∈Ω ρ (C 0 ) x∗ ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 13/29

**Accuracy** **Function**ProblemFormulationStability and**Accuracy****Function**sIn a similar way, for a given δ ∈ [0, 1), we consider a setΘ δ (C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| ≤ δ · c 0 ij, i ∈ N m , j ∈ N m }.For a J-equilibrium x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the valueof the accuracy function is defined as follows:A(C 0 , J, x ∗ , δ) :=max ε(C, J,C∈Θ δ (C 0 ) x∗ ).It is easy to check that S(C 0 , J, x ∗ , ρ) ≥ 0 for anyρ ∈ [0, q(C 0 , X)) as well as A(C 0 , J, x ∗ , δ) ≥ 0 for eachδ ∈ [0, 1).Sensitivity Analysis in Game Theory Yury Nikulin - p. 14/29

SufficiencyProblemFormulationStability and**Accuracy****Function**sDenoteΩ ′ ρ(C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| < ρ, i ∈ N m , j ∈ N m }.Proposition 1. For x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), we havex ∗ ∈ JE m (C, X) for any C ∈ Ω ′ ρ(C 0 ) if and only ifS(C 0 , J, x ∗ , ρ) = 0.DenoteΘ ′ δ(C 0 ) := {C ∈ R m×m+ : |c ij − c 0 ij| < δ · c 0 ij, i ∈ N m , j ∈ N m }.Proposition 2. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), we havex ∗ ∈ JE m (C, X) for any C ∈ Θ ′ δ (C0 ) if and only ifA(C 0 , J, x ∗ , δ) = 0.Sensitivity Analysis in Game Theory Yury Nikulin - p. 15/29

Stability and **Accuracy** RadiiProblemFormulationStability and**Accuracy****Function**sFormally, the stability radius R S (C 0 , J, x ∗ ) and the accuracyradius R A (C 0 , J, x ∗ ) are defined in the following way:R S (C 0 , J, x ∗ ) := sup{ρ ∈ [0, q(C 0 , X)) : S(C 0 , J, x ∗ , ρ) = 0},R A (C 0 , J, x ∗ ) := sup{δ ∈ [0, 1) : A(C 0 , J, x ∗ , δ) = 0}.Sensitivity Analysis in Game Theory Yury Nikulin - p. 16/29

Formula for Stability **Function**ProblemFormulationStability and**Accuracy****Function**sTheorem 1. For x ∗ ∈ JE m (C 0 , X) and ρ ∈ [0, q(C 0 , X)), the stabilityfunction can be expressed by the formula:S(C 0 , J, x ∗ , ρ) = maxr∈N smax min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1.(1)Sensitivity Analysis in Game Theory Yury Nikulin - p. 17/29

Sketch of the proofProblemFormulationStability and**Accuracy****Function**sFirst we will prove that S(C 0 , J, x ∗ , ρ) ≤ Γ(C 0 , J, x ∗ , ρ), whereΓ(C 0 , J, x ∗ , ρ) is the right-hand side of (1). Indeed, changingthe order of minimums and maximums, we getS(C 0 , J, x ∗ , ρ) =max ε(C, J,C∈Ω ρ (C 0 ) x∗ ) =maxC∈Ω ρ (C 0 ) maxr∈N smaxr∈N smax min p i (C, x ∗ ) − p i (C, x)x∈W J r (x∗ ) i∈J r p i (C, x)max minx∈W J r (x∗ ) i∈J rmaxC∈Ω ρ (C 0 )C i (x ∗ − x)C i x.≤Sensitivity Analysis in Game Theory Yury Nikulin - p. 18/29

Sketch of the proofProblemFormulationStability and**Accuracy****Function**sFor any fixed r ∈ N s and x ∈ W Jr (x ∗ ) and all i ∈ J r themaximum of C i(x ∗ −x)C i xover C ∈ Ω ρ (C 0 ) is attained whenc ∗ ij =⎧⎪⎨⎪⎩c 0 ij − φ, if x∗ j = 0, i ∈ J r, j ∈ N m ;c 0 ij + φ, if x∗ j = 1, i ∈ J r, j ∈ N m ;c 0 ij , if i ∉ J r, j ∈ N m ,where φ ≥ 0 is taken to satisfy C ∗ ∈ Ω ρ (C 0 ). Thenmaxr∈N smaxx∈W J r (x∗ ) mini∈J rC ∗ i (x∗ − x)C ∗ i x =(2)maxr∈N smaxx∈W J r (x∗ ) mini∈J rC 0 i (x∗ − x) + ρ ‖ x ∗ − x ‖ 1C 0 i x − ρ ‖ x ‖ 1= Γ(C 0 , J, x ∗ , ρ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 19/29

Sketch of the proofProblemFormulationStability and**Accuracy****Function**sSo, we have that S(C 0 , J, x ∗ , ρ) ≤ Γ(C 0 , J, x ∗ , ρ). Now itremains to show that S(C 0 , J, x ∗ , ρ) ≥ Γ(C 0 , J, x ∗ , ρ). Considera matrix C ∗ with elements defined according to (2) for anyr ∈ N s . ThenS(C 0 , x ∗ , J, ρ) =maxC∈Ω ρ (C 0 ) ε(C, J, x∗ ) ≥ ε(C ∗ , J, x ∗ ) =maxr∈N smaxr∈N smaxx∈W J r (x∗ ) mini∈J rC ∗ i (x∗ − x)C ∗ i x =max min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1So, we have that S(C 0 , x ∗ , J, ρ) ≥ Γ(C 0 , x ∗ , J, ρ). Thiscompletes the proof.= Γ(C 0 , J, x ∗ , ρ).Sensitivity Analysis in Game Theory Yury Nikulin - p. 20/29

Formula for Stability RadiusProblemFormulationStability and**Accuracy****Function**sTheorem 2. For x ∗ ∈ JE m (C 0 , X), the stability radius can be expressedby the formula:R S (C 0 , J, x ∗ ) = min{q(C 0 , X), minr∈N smin max Ci 0(x∗ − x)}.x∈W J r (x∗ )\{x ∗ } i∈J r ‖ x ∗ − x ‖ 1(3)Sensitivity Analysis in Game Theory Yury Nikulin - p. 21/29

Sketch of the proofProblemFormulationStability and**Accuracy****Function**sIf ρ = 0, then S(C 0 , J, x ∗ , 0) = 0. Assume thatS(C 0 , J, x ∗ , ρ) > 0. Using formula (1) specified by theorem 1,we derive that S(C 0 , J, x ∗ , ρ) > 0 if and onlymaxr∈N smax min Ci 0(x∗ − x) + ρ ‖ x ∗ − x ‖ 1x∈W J r (x∗ ) i∈J r Ci 0x − ρ ‖ x ‖ 1> 0.Last inequality holds if and only ifρ ≥ ˜ρ := minr∈N sminx∈W J r (x∗ )\{x ∗ } maxi∈J rC 0 i (x − x∗ )‖ x − x ∗ ‖ 1.Thus, if ˜ρ ≤ q(C 0 , X), then we get that S(C 0 , J, x ∗ , ρ) = 0 onthe interval [0, ˜ρ). Otherwise the stability function is equal tozero on [0, q(C 0 , X)). This ends the proof.Sensitivity Analysis in Game Theory Yury Nikulin - p. 22/29

Independent players, no bansProblemFormulationStability and**Accuracy****Function**sFrom theorem 2, we get the following resultsCorollary 1. For a game with m independent players and no bans(J = ({1}, ..., {m}), B r = ∅ for all r ∈ N m ) the stability radius of aNash equilibrium x ∗ ∈ NE m (C 0 , X) can be expressed by the formulaR S (C 0 , J, x ∗ ) = mini∈N mc 0 ii.In other words, corollary 1 states that the stability radius ofx ∗ ∈ NE m (C 0 , X) is defined by elements on the principaldiagonal of C 0 .Sensitivity Analysis in Game Theory Yury Nikulin - p. 23/29

Independent players, no bansProblemFormulationStability and**Accuracy****Function**sNotice that, if the game does not accept bans, then the valueof stability (accuracy) radius is determined by principaldiagonal elements, i.e. R S (C 0 , J, x ∗ ) = q(C 0 , X)(R A (C 0 , J, x ∗ ) = 1). It means that in this case the stability andaccuracy functions cannot provide any additional informationabout game solution stability but only duplicate informationderived by means of stability and accuracy radii. That suggeststo use stability and accuracy functions as efficient tools ofstability analysis only for the games with numerous bans.Sensitivity Analysis in Game Theory Yury Nikulin - p. 24/29

**Accuracy** **Function** and RadiusProblemFormulationStability and**Accuracy****Function**sTheorem 3. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the accuracyfunction can be expressed by the formula:A(C 0 , J, x ∗ , δ) = maxr∈N smax minx∈W J r (x∗ )Ci 0(x∗ − x) + δ ∑ c 0 ij |x∗ j − x j|j∈N mi∈J r (1 − δ)Ci 0x .Theorem 4. For x ∗ ∈ JE m (C 0 , X) and δ ∈ [0, 1), the accuracy radiuscan be expressed by the formula:R A (C 0 , J, x ∗ , δ) = min{1, minr∈N sminx∈W J r (x∗ )\{x ∗ } maxCi 0(x∗ − x)∑i∈J r c 0 ij |x∗ j − x j|j∈N m}.Sensitivity Analysis in Game Theory Yury Nikulin - p. 25/29

ExampleProblemFormulationStability and**Accuracy****Function**sLet m = 3, s = 2, J = ({1}, {2, 3}) and⎛1 2 1C 0 ⎜= ⎝ 2 1 21 3 2⎞⎟⎠.Assume also that B = (∅, {(0, 0) T , (1, 1) T }), i.e.X = {x 1 , x 2 , x 3 , x 4 }, x 1 = (0, 0, 1) T , x 2 = (0, 1, 0) T ,x 3 = (1, 0, 1) T , and x 4 = (1, 1, 0) T .Then C 0 x 1 = (1, 2, 2) T , C 0 x 2 = (2, 1, 3) T , C 0 x 3 = (2, 4, 3) T ,C 0 x 4 = (3, 3, 4) T , and JE 3 (C 0 , X) = {x 1 , x 2 }. Using formula(3), we calculate R 3 (C 0 , J, x 1 ) = 1 2 and R3 (C 0 , J, x 2 ) = 1 2 . Ithappens that for both solutions x 1 and x 2 , the stability radius isequal to 1 2 . To get more information about x1 and x 2 , we maycheck the behavior of these solutions when they becomenon-equilibria by means of calculating stability functions oninterval [0, q(C 0 , X)), q(C 0 , X) = 1 using formula (1):Sensitivity Analysis in Game Theory Yury Nikulin - p. 26/29

ExampleProblemFormulationStability and**Accuracy****Function**s{S(C 0 , J, x 1 , ρ) = max{S(C 0 , J, x 2 , ρ) = maxmax { 0, −1 + ρ2 − 2ρmax{0, −1 + ρ3 − 2ρ}, max{0, min{1 + 2ρ1 − ρ}, max{0,−1 + 2ρ2 − ρ } },,−1 + 2ρ3 − ρwhere ρ ∈ [0, 1).Sensitivity Analysis in Game Theory Yury Nikulin - p. 27/29

ExampleProblemFormulation1.0Stability and**Accuracy****Function**s0.750.50.250.00.00.250.5rho0.751.0Figure 1: Stability functions S(C 0 , J, x 1 , ρ) and S(C 0 , J, x 2 , ρ)for ρ ∈ [0, 1).Graphics for S(C 0 , J, x 1 , ρ) (continuous line) and S(C 0 , J, x 2 , ρ)(dotted line) are depicted in figure 1.Sensitivity Analysis in Game Theory Yury Nikulin - p. 28/29

Concluding remarksProblemFormulationStability and**Accuracy****Function**sThe accuracy and stability functions describe the quality ofequilibrium in the game with uncertain coefficients in payoffs.The definitions of these functions are directly connected withgiven optimality principles. While the stability and accuracyradii determine only maximum values of independentperturbations preserving the property of being equilibrium for agiven solution, the stability and accuracy functions willadditionally give the answer what happens with the solutionafter it becomes non-optimal in terms of quantitative measureof stability reflecting the speed of relative deterioration of thegiven solution.Sensitivity Analysis in Game Theory Yury Nikulin - p. 29/29