Signature
NE_11 = (u_11 >= u_21) and (v_11 >= v_12); NE_12 = (u_12 >= u_22) and (v_12 >= v_11); NE_21 = (u_21 >= u_11) and (v_21 >= v_22); NE_22 = (u_22 >= u_12) and (v_22 >= v_21); S_11 = (u_11 > u_21) and (v_11 > v_12); S_12 = (u_12 > u_22) and (v_12 > v_11); S_21 = (u_21 > u_11) and (v_21 > v_22); S_22 = (u_22 > u_12) and (v_22 > v_21)
| Inputs | Definition | Unit |
|---|---|---|
u_11 | Payoff to the row player when it plays its first strategy and the column player plays its first | the row player's payoff unit, such as GBP million a year |
u_21 | Payoff to the row player when it plays its second strategy and the column player plays its first | the row player's payoff unit |
v_11 | Payoff to the column player when the row player plays its first strategy and the column player plays its first | the column player's payoff unit, such as GBP million a year |
v_12 | Payoff to the column player when the row player plays its first strategy and the column player plays its second | the column player's payoff unit |
u_12 | Payoff to the row player when it plays its first strategy and the column player plays its second | the row player's payoff unit |
u_22 | Payoff to the row player when it plays its second strategy and the column player plays its second | the row player's payoff unit |
v_21 | Payoff to the column player when the row player plays its second strategy and the column player plays its first | the column player's payoff unit |
v_22 | Payoff to the column player when the row player plays its second strategy and the column player plays its second | the column player's payoff unit |
NE_11 | 1 when the cell pairing the row player's first strategy with the column player's first passes both best-reply comparisons with ties allowed, otherwise 0 | indicator (1 or 0) |
|---|---|---|
NE_12 | 1 when the cell pairing the row player's first strategy with the column player's second passes both best-reply comparisons with ties allowed, otherwise 0 | indicator (1 or 0) |
NE_21 | 1 when the cell pairing the row player's second strategy with the column player's first passes both best-reply comparisons with ties allowed, otherwise 0 | indicator (1 or 0) |
NE_22 | 1 when the cell pairing the row player's second strategy with the column player's second passes both best-reply comparisons with ties allowed, otherwise 0 | indicator (1 or 0) |
S_11 | 1 when the cell pairing the row player's first strategy with the column player's first passes both best-reply comparisons with strict inequalities, otherwise 0 | indicator (1 or 0) |
S_12 | 1 when the cell pairing the row player's first strategy with the column player's second passes both best-reply comparisons with strict inequalities, otherwise 0 | indicator (1 or 0) |
S_21 | 1 when the cell pairing the row player's second strategy with the column player's first passes both best-reply comparisons with strict inequalities, otherwise 0 | indicator (1 or 0) |
S_22 | 1 when the cell pairing the row player's second strategy with the column player's second passes both best-reply comparisons with strict inequalities, otherwise 0 | indicator (1 or 0) |
Function
Nash equilibria of a two-player game with two strategies each
Maps the payoffs of two players, each with two pure strategies, over the four strategy pairs to the game's Nash equilibria, the pairs in which each player's strategy is a best reply to the other's, u_i(s_i*, s_-i*) >= u_i(s_i, s_-i*) for every strategy s_i of every player i, as in the article's display. The general condition over any finite strategy sets has no calculator form. For a 2 x 2 game HE-FM-NASH-001 tests each of the four cells as a pure-strategy equilibrium and classifies it as strict or weak, HE-FM-NASH-002 gives the mixed equilibrium in which each player randomises so as to leave the other indifferent, and HE-CF-NASH-001 lists every equilibrium of a game without payoff ties and values the outcome of mixed play. Each player's expected payoff at the mixed equilibrium and the expected value of an outcome measure such as total surplus are probability-weighted sums, the form of HE-FM-CHN-001, and the probability of each cell under independent mixing is the product of HE-FM-CHN-002, so neither is restated. Dominance between interventions in cost-effectiveness analysis (HE-FM-DOM-001) compares the incremental cost and effect of options for one decision maker and is a different test. The Nash bargaining solution (HE-FN-BARG-001, with the negotiated price of HE-FM-BARG-001) divides a surplus between parties who can make a binding agreement, Rubinstein's first proposer's share (HE-FM-BARG-003) comes from the perfect equilibrium of a sequential alternating-offers game, and the voluntary vaccination coverage of HE-FM-VEXT-003 is the equilibrium of a population game solved through an infection risk; all are separate records. Notation follows the Nash Equilibrium article, with u_jk and v_jk for the row and column players' payoffs when the row player chooses its strategy j and the column player its strategy k.
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Implementations
Excel
Excel best-reply check of the four cells of a 2 x 2 game
The row player's first and second strategies are Up and Down and the column player's are Left and Right. With the row player's payoffs in cells named PayoffAUpLeft, PayoffAUpRight, PayoffADownLeft and PayoffADownRight and the column player's in PayoffBUpLeft, PayoffBUpRight, PayoffBDownLeft and PayoffBDownRight, the first four formulas go in EquilibriumUpLeft, EquilibriumUpRight, EquilibriumDownLeft and EquilibriumDownRight and the last four in StrictUpLeft, StrictUpRight, StrictDownLeft and StrictDownRight. Each returns TRUE or FALSE, which Excel counts as 1 or 0 in arithmetic.
=AND(PayoffAUpLeft>=PayoffADownLeft,PayoffBUpLeft>=PayoffBUpRight); =AND(PayoffAUpRight>=PayoffADownRight,PayoffBUpRight>=PayoffBUpLeft); =AND(PayoffADownLeft>=PayoffAUpLeft,PayoffBDownLeft>=PayoffBDownRight); =AND(PayoffADownRight>=PayoffAUpRight,PayoffBDownRight>=PayoffBDownLeft); =AND(PayoffAUpLeft>PayoffADownLeft,PayoffBUpLeft>PayoffBUpRight); =AND(PayoffAUpRight>PayoffADownRight,PayoffBUpRight>PayoffBUpLeft); =AND(PayoffADownLeft>PayoffAUpLeft,PayoffBDownLeft>PayoffBDownRight); =AND(PayoffADownRight>PayoffAUpRight,PayoffBDownRight>PayoffBDownLeft)
Assumptions
One-shot simultaneous game with two players and two strategies each
The two players choose at the same time without seeing each other's choice, each knows both players' payoffs, and the payoffs are those of a game in matrix form, as in section 5.1.1 of Tadelis. When one player sees the other's choice before moving, the game is sequential and is solved by backward induction under the Game Theory concept, as in step 5 of the article's example.
Each player's payoffs compared only with that player's own payoffs
The row player's comparisons use only u_jk and the column player's only v_jk, so the two players' payoffs need not share a unit. The check says nothing about which cell is best for both players or for the health system, which needs a separate valuation of each cell, as in step 4 of the article's example.
Ties count as best replies in the weak equilibrium test
Definition 5.1 of Tadelis requires the equilibrium payoff to be at least the payoff from any alternative, so a player that does exactly as well with either strategy against the other's choice is playing a best reply with both, and the cell passes NE_jk. Definition 7 of Slantchev uses a strict inequality, so the same cell fails S_jk and is a weak equilibrium. The tie rule of cost-effectiveness dominance (HE-FM-DOM-001) is a different rule and is not used here.
Worked examples
Two hospitals deciding on one angioplasty service have two strict pure equilibria
The article's illustrative game, with Invest as each hospital's first strategy and Stay out as its second, payoffs in GBP million a year. (Invest, Stay out) and (Stay out, Invest) pass both comparisons strictly. (Invest, Invest) fails because each hospital gains 2 by staying out, and (Stay out, Stay out) fails because each gains 4 by investing, as in step 1 of the article.
u_11 = -2; u_12 = 4; u_21 = 0; u_22 = 0; v_11 = -2; v_12 = 0; v_21 = 4; v_22 = 0; NE_11 = 0; NE_12 = 1; NE_21 = 1; NE_22 = 0; S_11 = 0; S_12 = 1; S_21 = 1; S_22 = 0
Angioplasty game in which duplication breaks even has three weak pure equilibria
Step 6 of the article, with each hospital earning 0 in place of a loss of 2 when both invest. The three cells with at least one investor pass the weak test, and none passes the strict test, because in each of them one hospital does exactly as well by switching.
u_11 = 0; u_12 = 4; u_21 = 0; u_22 = 0; v_11 = 0; v_12 = 0; v_21 = 4; v_22 = 0; NE_11 = 1; NE_12 = 1; NE_21 = 1; NE_22 = 0; S_11 = 0; S_12 = 0; S_21 = 0; S_22 = 0
Matching pennies has no pure-strategy equilibrium
Slantchev's matching pennies game (Figure 18), in which the row player wins 1 when the two choices match and the column player wins 1 when they differ. In every cell one player gains 2 by switching, so no cell passes. The game's only equilibrium is mixed (HE-EX-NASH-007).
u_11 = 1; u_12 = -1; u_21 = -1; u_22 = 1; v_11 = -1; v_12 = 1; v_21 = 1; v_22 = -1; NE_11 = 0; NE_12 = 0; NE_21 = 0; NE_22 = 0; S_11 = 0; S_12 = 0; S_21 = 0; S_22 = 0
Hi-lo coordination game has two strict pure equilibria
The Hi-lo game the article takes from the Stanford Encyclopedia of Philosophy, in which matching on the first option pays 10 each, matching on the second pays 1 each and mismatching pays nothing. Both matched cells pass strictly, so the test alone cannot say which will occur.
u_11 = 10; u_12 = 0; u_21 = 0; u_22 = 1; v_11 = 10; v_12 = 0; v_21 = 0; v_22 = 1; NE_11 = 1; NE_12 = 0; NE_21 = 0; NE_22 = 1; S_11 = 1; S_12 = 0; S_21 = 0; S_22 = 1
Common errors
Checking only one player's best reply in a 2 x 2 game
A cell is an equilibrium only when both players are playing a best reply. Testing the row player alone in matching pennies marks the two matched cells as equilibria, although in each of them the column player moves from minus 1 to 1 by switching (computed here), and the game has no pure equilibrium.
Strict inequality used in the weak best-reply test
Writing the weak test with > in place of >= drops every equilibrium that rests on a tie. In the break-even angioplasty game of HE-EX-NASH-002 it finds no pure equilibrium where there are three (computed here), and the article's selection of (Invest, Invest) by removing the weakly dominated choice then has nothing to select from.
Reading an equilibrium cell as the best outcome
The check finds stable cells, not efficient ones, and Tadelis states that Nash equilibrium does not guarantee Pareto optimality. In the Hi-lo game of HE-EX-NASH-004 the second matched cell passes both the weak and the strict test although each player earns 1 there against 10 in the other equilibrium.
Sources
Tadelis on the pure-strategy Nash equilibrium and the matrix method
Tadelis S. Pinning down beliefs: Nash equilibrium. Chapter 5 in Game Theory: An Introduction. Princeton: Princeton University Press; 2013 (publisher's sample chapter, full text read). Section 5.1 (Nash Equilibrium in Pure Strategies), Definition 5.1, a pure-strategy profile is a Nash equilibrium when each player's strategy is a best response to the others', v_i(s_i*, s_-i*) at least v_i(s_i', s_-i*) for every alternative s_i' and every player i. Section 5.1.1 (Pure-Strategy Nash Equilibrium in a Matrix), the three-step method that marks the row player's best reply in each column and the column player's best reply in each row, a cell marked twice being a pure-strategy Nash equilibrium. Section 5.1.2 (Evaluating the Nash Equilibria Solution) states that Nash equilibrium does not guarantee Pareto optimality.
Slantchev on strict Nash equilibrium and matching pennies
Slantchev BL. Game theory: dominance, Nash equilibrium, symmetry. Lecture notes, Department of Political Science, University of California, San Diego; 23 May 2008 (full text read). Section 2.1 (Pure-Strategy Nash Equilibrium), Definition 4, a best response as a strategy with a payoff at least that of every alternative given the others' strategies. Section 2.2 (Strict Nash Equilibrium), Definition 7, a strict Nash equilibrium requires a strictly higher payoff than every other strategy, with the Battle of the Bismarck Sea as an equilibrium that is not strict because one player is indifferent. Section 2.3 (Mixed Strategy Nash Equilibrium), Figure 18, matching pennies has no Nash equilibrium in pure strategies.
Nash on equilibrium points as self-countering strategy combinations
Nash JF. Equilibrium points in n-person games. Proc Natl Acad Sci USA. 1950;36(1):48-49 (full text read from the original page scan). Defines a game with a finite set of pure strategies for each player and a payment to each player for each combination, one combination countering another when each player's strategy in it yields the highest obtainable expectation against the other players' strategies in the second, and an equilibrium point as a self-countering combination.
Canonical Identity
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