Cheatsheets — #equilibria

Quick-reference cards for the formal apparatus most often invoked across the tutorials: equilibrium concepts, payoff conventions, common solution methods.
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Cheatsheets

Compact, one-page references for the apparatus you reach for most often: equilibrium concepts, payoff notation, common solvers, R idioms.

Equilibrium concepts at a glance

Pure-strategy Nash equilibrium

A profile \(s^* = (s_1^*, \dots, s_n^*)\) where no unilateral deviation pays: \[u_i(s_i^*, s_{-i}^*) \ge u_i(s_i, s_{-i}^*) \quad \forall i, \; \forall s_i \in S_i.\] Existence not guaranteed in pure strategies for finite games.

Mixed-strategy Nash equilibrium

Each player randomises over \(S_i\) via a distribution \(\sigma_i\) such that every pure strategy in the support of \(\sigma_i^*\) yields the same expected payoff. Existence guaranteed for any finite normal-form game (Nash 1951).

Subgame-perfect equilibrium

A Nash equilibrium of the full game whose restriction to every subgame is also a Nash equilibrium. Found by backward induction in finite extensive-form games with perfect information.

Bayesian Nash equilibrium

For games of incomplete information: each player’s strategy maps from their type \(\theta_i\) to actions, and best-responds in expectation given beliefs over other players’ types.

Perfect Bayesian equilibrium

Refines subgame-perfection for extensive-form games of incomplete information: requires both sequential rationality at every information set and beliefs consistent with the strategy profile via Bayes’ rule wherever possible.

Correlated equilibrium

A probability distribution over the joint action space such that, conditional on the recommendation a player receives, deviating is not profitable. Superset of Nash; computationally easier (LP, not fixed-point).

Payoff notation conventions

Symbol Meaning
\(N\) Set of players, \(\|N\|=n\)
\(S_i\) Pure strategy set of player \(i\)
\(\Delta(S_i)\) Mixed strategies = probability simplex over \(S_i\)
\(u_i\) Utility/payoff function for player \(i\)
\(s, s_i\) Strategy profile, individual strategy
\(s_{-i}\) Profile of everyone except \(i\)
\(\sigma\) Mixed strategy profile
\(v(S)\) Characteristic function (coalition value) in cooperative games
\(\phi_i\) Shapley value of player \(i\)
\(\theta_i\) Player \(i\)’s type (incomplete information)

Solution-method picker

If the game is… Reach for…
Finite, normal-form, complete information Iterated elimination → pure NE; LP for zero-sum
2×2 finite Best-response correspondences; mixed-NE indifference equations
Sequential, perfect information Backward induction
Sequential, imperfect information Sequential equilibrium / PBE
Incomplete information Type space + Bayesian NE; for designer-set rules, mechanism design
Repeated, finite horizon Last-stage NE then back-induct
Repeated, infinite horizon Folk theorem; trigger strategies; abreu-pearce-stacchetti
Cooperative (TU) Shapley value, core, nucleolus
Cooperative (NTU) Nash bargaining solution
Evolutionary Replicator dynamics, ESS
Strategic learning agents Fictitious play, no-regret, multi-agent RL

For an interactive walk-through, see the Decision Assistant.

Common R idioms

# Best response in a 2-player bimatrix game
best_response <- function(payoffs, opp_strategy) {
  # payoffs: own payoff matrix (rows = own actions, cols = opp actions)
  # opp_strategy: probability vector over opp actions
  expected <- payoffs %*% opp_strategy
  which(expected == max(expected))
}
# Mixed-strategy NE in 2×2 via indifference
# Solve p such that opp is indifferent
solve_2x2_mixed <- function(A, B) {
  # A: row player's payoff matrix; B: column player's
  p <- (B[2,2] - B[2,1]) / (B[1,1] - B[1,2] - B[2,1] + B[2,2])
  q <- (A[2,2] - A[1,2]) / (A[1,1] - A[2,1] - A[1,2] + A[2,2])
  list(p = p, q = q)
}
# Shapley value of an n-player TU game via the permutation formula
shapley <- function(v, n) {
  # v: characteristic function as a named list indexed by coalition strings
  sapply(1:n, function(i) {
    others <- setdiff(1:n, i)
    sum(sapply(0:length(others), function(k) {
      sets <- combn(others, k, simplify = FALSE)
      sapply(sets, function(S) {
        S_key   <- paste(sort(S),     collapse = ",")
        Si_key  <- paste(sort(c(S, i)), collapse = ",")
        w <- factorial(k) * factorial(n - k - 1) / factorial(n)
        w * ((v[[Si_key]] %||% 0) - (v[[S_key]] %||% 0))
      }) |> unlist() |> sum()
    }))
  })
}

See also

  • Glossary — formal definitions, alphabetised.
  • Common errors — frequently miscalled traps in setting up and reading games.
  • Decision Assistant — interactive picker for the right model + solution concept.
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