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alpha_rank.py

Source:alpha_rank.py

...315    """Gets Markov transition matrix for multipopulation games."""316    num_strats_per_population = utils.get_num_strats_per_population(317        payoff_tables, payoffs_are_hpt_format318    )319    num_profiles = utils.get_num_profiles(num_strats_per_population)320    eta = 1.0 / (np.sum(num_strats_per_population - 1))321    c = np.zeros((num_profiles, num_profiles))322    rhos = np.zeros((num_profiles, num_profiles))323    for id_row_profile in range(num_profiles):324        row_profile = utils.get_strat_profile_from_id(325            num_strats_per_population, id_row_profile326        )327        next_profile_gen = utils.get_valid_next_profiles(328            num_strats_per_population, row_profile329        )330        for index_population_that_changed, col_profile in next_profile_gen:331            id_col_profile = utils.get_id_from_strat_profile(332                num_strats_per_population, col_profile333            )334            if use_inf_alpha:335                payoff_col = _get_payoff(336                    payoff_tables[index_population_that_changed],337                    payoffs_are_hpt_format,338                    col_profile,339                    k=index_population_that_changed,340                )341                payoff_row = _get_payoff(342                    payoff_tables[index_population_that_changed],343                    payoffs_are_hpt_format,344                    row_profile,345                    k=index_population_that_changed,346                )347                if np.isclose(payoff_col, payoff_row, atol=1e-14):348                    c[id_row_profile, id_col_profile] = eta * 0.5349                elif payoff_col > payoff_row:350                    # Transition to col strategy since its payoff is higher than row351                    # strategy, but remove some small amount of mass, inf_alpha_eps, to352                    # keep the chain irreducible353                    c[id_row_profile, id_col_profile] = eta * (1 - inf_alpha_eps)354                else:355                    # Transition with very small probability356                    c[id_row_profile, id_col_profile] = eta * inf_alpha_eps357            else:358                rhos[id_row_profile, id_col_profile] = _get_rho_sr_multipop(359                    payoff_table_k=payoff_tables[index_population_that_changed],360                    payoffs_are_hpt_format=payoffs_are_hpt_format,361                    k=index_population_that_changed,362                    m=m,363                    r=col_profile,364                    s=row_profile,365                    alpha=alpha,366                )367                c[id_row_profile, id_col_profile] = (368                    eta * rhos[id_row_profile, id_col_profile]369                )370        # Special case of self-transition371        c[id_row_profile, id_row_profile] = 1 - sum(c[id_row_profile, :])372    return c, rhos373def _get_stationary_distr(c):374    """Gets stationary distribution of transition matrix c."""375    eigenvals, left_eigenvecs, _ = la.eig(c, left=True, right=True)376    mask = abs(eigenvals - 1.0) < 1e-10377    left_eigenvecs = left_eigenvecs[:, mask]378    num_stationary_eigenvecs = np.shape(left_eigenvecs)[1]379    if num_stationary_eigenvecs != 1:380        raise ValueError(381            "Expected 1 stationary distribution, but found %d"382            % num_stationary_eigenvecs383        )384    left_eigenvecs *= 1.0 / sum(left_eigenvecs)385    return left_eigenvecs.real.flatten()386def print_results(387    payoff_tables, payoffs_are_hpt_format, rhos=None, rho_m=None, c=None, pi=None388):389    """Prints the finite-population analysis results."""390    print("Payoff tables:\n")391    if payoffs_are_hpt_format:392        for payoff_table in payoff_tables:393            print(payoff_table())394    else:395        print(payoff_tables)396    if rho_m is not None:397        print("\nNeutral fixation probability (rho_m):\n", rho_m)398    if rhos is not None and rho_m is not None:399        print(400            "\nFixation probability matrix (rho_{r,s}/rho_m):\n",401            np.around(rhos / rho_m, decimals=2),402        )403    if c is not None:404        print("\nMarkov transition matrix (c):\n", np.around(c, decimals=2))405    if pi is not None:406        print("\nStationary distribution (pi):\n", pi)407def sweep_pi_vs_epsilon(408    payoff_tables,409    strat_labels=None,410    warm_start_epsilon=None,411    visualize=False,412    return_epsilon=False,413    min_iters=10,414    max_iters=100,415    min_epsilon=1e-14,416    num_strats_to_label=10,417    legend_sort_clusters=False,418):419    """Computes infinite-alpha distribution for a range of perturbations.420    The range of response graph perturbations is defined in epsilon_list.421    Note that min_iters and max_iters is necessary as it may sometimes appear the422    stationary distribution has converged for a game in the first few iterations,423    where in reality a sufficiently smaller epsilon is needed for the distribution424    to first diverge, then reconverge. This behavior is dependent on both the425    payoff structure and bounds, so the parameters min_iters and max_iters can be426    used to fine-tune this.427    Args:428      payoff_tables: List of game payoff tables, one for each agent identity.429        Each payoff_table may be either a numpy array, or a430        _PayoffTableInterface object.431      strat_labels: Human-readable strategy labels. See get_strat_profile_labels()432        in utils.py for formatting details.433      warm_start_epsilon: Initial value of epsilon to use.434      visualize: Plot the sweep results.435      return_epsilon: Whether to return the final epsilon used.436      min_iters: the minimum number of sweep iterations.437      max_iters: the maximum number of sweep iterations.438      min_epsilon: the minimum value of epsilon to be tested, at which point the439        sweep terminates (if not converged already).440      num_strats_to_label: Number of strats to label in legend441      legend_sort_clusters: If true, strategies in the same cluster are sorted in442        the legend according to orderings for earlier alpha values. Primarily for443        visualization purposes! Rankings for lower alpha values should be444        interpreted carefully.445    Returns:446     pi: AlphaRank stationary distribution.447     epsilon: The AlphaRank transition matrix noise level resulting from sweep.448    """449    payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)450    num_populations = len(payoff_tables)451    num_strats_per_population = utils.get_num_strats_per_population(452        payoff_tables, payoffs_are_hpt_format453    )454    if num_populations == 1:455        num_profiles = num_strats_per_population[0]456    else:457        num_profiles = utils.get_num_profiles(num_strats_per_population)458    assert (459        strat_labels is None460        or isinstance(strat_labels, dict)461        or (len(strat_labels) == num_profiles)462    )463    pi_list = np.empty((num_profiles, 0))464    pi, alpha, m = None, None, None  # Unused in infinite-alpha regime465    epsilon_list = []466    epsilon_pi_hist = {}467    num_iters = 0468    epsilon_mult_factor = 0.5469    alpharank_succeeded_once = False470    if warm_start_epsilon is not None:471        epsilon = warm_start_epsilon472    else:473        epsilon = 0.5474    while True:475        try:476            pi_prev = pi477            _, _, pi, _, _ = compute(478                payoff_tables,479                m=m,480                alpha=alpha,481                use_inf_alpha=True,482                inf_alpha_eps=epsilon,483            )484            epsilon_pi_hist[epsilon] = pi485            # Stop when pi converges486            if num_iters > min_iters and np.allclose(pi, pi_prev):487                break488            epsilon *= epsilon_mult_factor489            num_iters += 1490            alpharank_succeeded_once = True491            assert num_iters < max_iters, (492                "Alpharank stationary distr. not found"493                "after {} iterations of pi_vs_epsilon"494                "sweep".format(num_iters)495            )496        except ValueError as _:497            print("Error: ", _, epsilon, min_epsilon)498            # Case where epsilon has been decreased beyond desirable limits but no499            # distribution found.500            assert epsilon >= min_epsilon, (501                "AlphaRank stationary distr. not found &" "epsilon < min_epsilon."502            )503            # Case where epsilon >= min_epsilon, but still small enough that it causes504            # causes exceptions due to precision issues. So increase it.505            epsilon /= epsilon_mult_factor506            # Case where alpharank_succeeded_once (i.e., epsilon_list and pi_list have507            # at least one entry), and a) has not converged yet and b) failed on this508            # instance due to epsilon being too small. I.e., the rate of decreasing509            # of epsilon is too high.510            if alpharank_succeeded_once:511                epsilon_mult_factor = (epsilon_mult_factor + 1.0) / 2.0512                epsilon *= epsilon_mult_factor513    epsilon_list, pi_list = zip(514        *[515            (epsilon, epsilon_pi_hist[epsilon])516            for epsilon in sorted(epsilon_pi_hist.keys(), reverse=True)517        ]518    )519    pi_list = np.asarray(pi_list)520    if visualize:521        if strat_labels is None:522            strat_labels = utils.get_strat_profile_labels(523                payoff_tables, payoffs_are_hpt_format524            )525        alpharank_visualizer.plot_pi_vs_alpha(526            pi_list.T,527            epsilon_list,528            num_populations,529            num_strats_per_population,530            strat_labels,531            num_strats_to_label=num_strats_to_label,532            legend_sort_clusters=legend_sort_clusters,533            xlabel=r"Infinite-AlphaRank Noise $\epsilon$",534        )535    if return_epsilon:536        return pi_list[-1], epsilon_list[-1]537    else:538        return pi_list[-1]539def sweep_pi_vs_alpha(540    payoff_tables,541    strat_labels=None,542    warm_start_alpha=None,543    visualize=False,544    return_alpha=False,545    m=50,546    rtol=1e-5,547    atol=1e-8,548    num_strats_to_label=10,549    legend_sort_clusters=False,550):551    """Computes stationary distribution, pi, for range of selection intensities.552    The range of selection intensities is defined in alpha_list and corresponds553    to the temperature of the Fermi selection function.554    Args:555      payoff_tables: List of game payoff tables, one for each agent identity. Each556        payoff_table may be either a numpy array, or a _PayoffTableInterface557        object.558      strat_labels: Human-readable strategy labels. See get_strat_profile_labels()559        in utils.py for formatting details.560      warm_start_alpha: Initial value of alpha to use.561      visualize: Plot the sweep results.562      return_alpha: Whether to return the final alpha used.563      m: AlphaRank population size.564      rtol: The relative tolerance parameter for np.allclose calls.565      atol: The absolute tolerance parameter for np.allclose calls.566      num_strats_to_label: Number of strats to label in legend567      legend_sort_clusters: If true, strategies in the same cluster are sorted in568        the legend according to orderings for earlier alpha values. Primarily for569        visualization purposes! Rankings for lower alpha values should be570        interpreted carefully.571    Returns:572     pi: AlphaRank stationary distribution.573     alpha: The AlphaRank selection-intensity level resulting from sweep.574    """575    payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)576    num_populations = len(payoff_tables)577    num_strats_per_population = utils.get_num_strats_per_population(578        payoff_tables, payoffs_are_hpt_format579    )580    if num_populations == 1:581        num_profiles = num_strats_per_population[0]582    else:583        num_profiles = utils.get_num_profiles(num_strats_per_population)584    assert (585        strat_labels is None586        or isinstance(strat_labels, dict)587        or (len(strat_labels) == num_profiles)588    )589    pi_list = np.empty((num_profiles, 0))590    alpha_list = []591    num_iters = 0592    alpha_mult_factor = 2.0593    if warm_start_alpha is not None:594        alpha = warm_start_alpha595        alpharank_succeeded_once = False596    else:597        alpha = 1e-4  # Reasonable default for most games, can be user-overridden598    while 1:599        try:600            _, _, pi, _, _ = compute(payoff_tables, alpha=alpha, m=m)601            pi_list = np.append(pi_list, np.reshape(pi, (-1, 1)), axis=1)602            alpha_list.append(alpha)603            # Stop when pi converges604            if num_iters > 0 and np.allclose(pi, pi_list[:, num_iters - 1], rtol, atol):605                break606            alpha *= alpha_mult_factor607            num_iters += 1608            alpharank_succeeded_once = True609        except ValueError as _:610            if warm_start_alpha is not None and not alpharank_succeeded_once:611                # When warm_start_alpha is used, there's a chance that612                # the initial warm_start_alpha is too large and causes exceptions due to613                # the Markov transition matrix being reducible. So keep decreasing until614                # a single success occurs.615                alpha /= 2616            elif not np.allclose(pi_list[:, -1], pi_list[:, -2], rtol, atol):617                # Sweep stopped due to multiple stationary distributions, but pi had618                # not converged due to the alpha scaling being too large.619                alpha /= alpha_mult_factor620                alpha_mult_factor = (alpha_mult_factor + 1.0) / 2.0621                alpha *= alpha_mult_factor622            else:623                break624    if visualize:625        if strat_labels is None:626            strat_labels = utils.get_strat_profile_labels(627                payoff_tables, payoffs_are_hpt_format628            )629        alpharank_visualizer.plot_pi_vs_alpha(630            pi_list.T,631            alpha_list,632            num_populations,633            num_strats_per_population,634            strat_labels,635            num_strats_to_label=num_strats_to_label,636            legend_sort_clusters=legend_sort_clusters,637        )638    if return_alpha:639        return pi, alpha640    else:641        return pi642def compute_and_report_alpharank(643    payoff_tables, m=50, alpha=100, verbose=False, num_top_strats_to_print=8644):645    """Computes and visualizes Alpha-Rank outputs.646    Args:647      payoff_tables: List of game payoff tables, one for each agent identity. Each648        payoff_table may be either a numpy array, or a _PayoffTableInterface649        object.650      m: Finite population size.651      alpha: Fermi distribution temperature parameter.652      verbose: Set to True to print intermediate results.653      num_top_strats_to_print: Number of top strategies to print.654    Returns:655      pi: AlphaRank stationary distribution/rankings.656    """657    payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)658    rhos, rho_m, pi, _, _ = compute(payoff_tables, m=m, alpha=alpha)659    strat_labels = utils.get_strat_profile_labels(payoff_tables, payoffs_are_hpt_format)660    if verbose:661        print_results(payoff_tables, payoffs_are_hpt_format, pi=pi)662        utils.print_rankings_table(663            payoff_tables,664            pi,665            strat_labels,666            num_top_strats_to_print=num_top_strats_to_print,667        )668        m_network_plotter = alpharank_visualizer.NetworkPlot(669            payoff_tables, rhos, rho_m, pi, strat_labels, num_top_profiles=8670        )671        m_network_plotter.compute_and_draw_network()672    # print(strat_labels)673    return pi674def compute(675    payoff_tables,676    m=50,677    alpha=100,678    use_local_selection_model=True,679    verbose=False,680    use_inf_alpha=False,681    inf_alpha_eps=0.01,682):683    """Computes the finite population stationary statistics.684    Args:685      payoff_tables: List of game payoff tables, one for each agent identity. Each686        payoff_table may be either a numpy array, or a _PayoffTableInterface687        object.688      m: Finite population size.689      alpha: Fermi distribution temperature parameter.690      use_local_selection_model: Enable local evolutionary selection model, which691        considers fitness against the current opponent only, rather than the692        global population state.693      verbose: Set to True to print intermediate results.694      use_inf_alpha: Use infinite-alpha alpharank model.695      inf_alpha_eps: Noise term to use in infinite-alpha alpharank model.696    Returns:697      rhos: Matrix of strategy-to-strategy fixation probabilities.698      rho_m: Neutral fixation probability.699      pi: Finite population stationary distribution.700      num_strats: Number of available strategies.701    """702    payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)703    num_populations = len(payoff_tables)704    num_strats_per_population = utils.get_num_strats_per_population(705        payoff_tables, payoffs_are_hpt_format706    )707    # Handles the trivial case of Markov chain with one state708    if np.array_equal(709        num_strats_per_population, np.ones(len(num_strats_per_population))710    ):711        rhos = np.asarray([[1]])712        rho_m = 1.0 / m if not use_inf_alpha else 1713        num_profiles = 1714        pi = np.asarray([1.0])715        return rhos, rho_m, pi, num_profiles, num_strats_per_population716    if verbose:717        print("Constructing c matrix")718        print("num_strats_per_population:", num_strats_per_population)719    if num_populations == 1:720        # User fast closed-form analysis for constant-sum single-population games721        game_is_constant_sum, payoff_sum = utils.check_is_constant_sum(722            payoff_tables[0], payoffs_are_hpt_format723        )724        if verbose:725            print(726                "game_is_constant_sum:",727                game_is_constant_sum,728                "payoff sum: ",729                payoff_sum,730            )731        # Single-population/symmetric game just uses the first player's payoffs732        c, rhos = _get_singlepop_transition_matrix(733            payoff_tables[0],734            payoffs_are_hpt_format,735            m,736            alpha,737            game_is_constant_sum,738            use_local_selection_model,739            payoff_sum,740            use_inf_alpha=use_inf_alpha,741            inf_alpha_eps=inf_alpha_eps,742        )743        num_profiles = num_strats_per_population[0]744    else:745        c, rhos = _get_multipop_transition_matrix(746            payoff_tables,747            payoffs_are_hpt_format,748            m,749            alpha,750            use_inf_alpha=use_inf_alpha,751            inf_alpha_eps=inf_alpha_eps,752        )753        num_profiles = utils.get_num_profiles(num_strats_per_population)754    pi = _get_stationary_distr(c)755    rho_m = 1.0 / m if not use_inf_alpha else 1  # Neutral fixation probability756    if verbose:757        print_results(payoff_tables, payoffs_are_hpt_format, rhos, rho_m, c, pi)758    return rhos, rho_m, pi, num_profiles, num_strats_per_population759def suggest_alpha(payoff_tables, tol=0.1):760    """Suggests an alpha for use in alpha-rank.761    The suggested alpha is approximately the smallest possible alpha such that762    the ranking has 'settled out'. It is calculated as763    -ln(tol)/min_gap_between_payoffs.764    The logic behind this settling out is that the fixation probabilities can be765    expanded as a series, and the relative size of each term in this series766    changes with alpha. As alpha gets larger and larger, one of the terms in767    this series comes to dominate, and this causes the ranking to settle768    down. Just how fast this domination happens is easy to calculate, and this769    function uses it to estimate the alpha by which the ranking has settled.770    You can find further discussion at the PR:771    https://github.com/deepmind/open_spiel/pull/403772    Args:773      payoff_tables: List of game payoff tables, one for each agent identity. Each774        payoff_table may be either a numpy array, or a _PayoffTableInterface775        object.776      tol: the desired gap between the first and second terms in the fixation777        probability expansion. A smaller tolerance leads to a larger alpha, and778        a 'more settled out' ranking.779    Returns:780      A suggested alpha.781    """782    payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)783    num_strats_per_population = utils.get_num_strats_per_population(784        payoff_tables, payoffs_are_hpt_format785    )786    num_profiles = utils.get_num_profiles(num_strats_per_population)787    gap = np.inf788    for id_row_profile in range(num_profiles):789        row_profile = utils.get_strat_profile_from_id(790            num_strats_per_population, id_row_profile791        )792        next_profile_gen = utils.get_valid_next_profiles(793            num_strats_per_population, row_profile794        )795        for index_population_that_changed, col_profile in next_profile_gen:796            payoff_table_k = payoff_tables[index_population_that_changed]797            f_r = _get_payoff(798                payoff_table_k,799                payoffs_are_hpt_format,800                col_profile,...

alpharank.py

Source:alpharank.py

...287                                    inf_alpha_eps=0.1):288  """Gets Markov transition matrix for multipopulation games."""289  num_strats_per_population = utils.get_num_strats_per_population(290      payoff_tables, payoffs_are_hpt_format)291  num_profiles = utils.get_num_profiles(num_strats_per_population)292  eta = 1. / (np.sum(num_strats_per_population - 1))293  c = np.zeros((num_profiles, num_profiles))294  rhos = np.zeros((num_profiles, num_profiles))295  for id_row_profile in range(num_profiles):296    row_profile = utils.get_strat_profile_from_id(num_strats_per_population,297                                                  id_row_profile)298    next_profile_gen = utils.get_valid_next_profiles(num_strats_per_population,299                                                     row_profile)300    for index_population_that_changed, col_profile in next_profile_gen:301      id_col_profile = utils.get_id_from_strat_profile(302          num_strats_per_population, col_profile)303      if use_inf_alpha:304        payoff_col = _get_payoff(305            payoff_tables[index_population_that_changed],306            payoffs_are_hpt_format,307            col_profile,308            k=index_population_that_changed)309        payoff_row = _get_payoff(310            payoff_tables[index_population_that_changed],311            payoffs_are_hpt_format,312            row_profile,313            k=index_population_that_changed)314        if np.isclose(payoff_col, payoff_row, atol=1e-14):315          c[id_row_profile, id_col_profile] = eta * 0.5316        elif payoff_col > payoff_row:317          # Transition to col strategy since its payoff is higher than row318          # strategy, but remove some small amount of mass, inf_alpha_eps, to319          # keep the chain irreducible320          c[id_row_profile, id_col_profile] = eta * (1 - inf_alpha_eps)321        else:322          # Transition with very small probability323          c[id_row_profile, id_col_profile] = eta * inf_alpha_eps324      else:325        rhos[id_row_profile, id_col_profile] = _get_rho_sr_multipop(326            payoff_table_k=payoff_tables[index_population_that_changed],327            payoffs_are_hpt_format=payoffs_are_hpt_format,328            k=index_population_that_changed,329            m=m,330            r=col_profile,331            s=row_profile,332            alpha=alpha)333        c[id_row_profile,334          id_col_profile] = eta * rhos[id_row_profile, id_col_profile]335    # Special case of self-transition336    c[id_row_profile, id_row_profile] = 1 - sum(c[id_row_profile, :])337  return c, rhos338def _get_stationary_distr(c):339  """Gets stationary distribution of transition matrix c."""340  eigenvals, left_eigenvecs, _ = la.eig(c, left=True, right=True)341  mask = abs(eigenvals - 1.) < 1e-10342  left_eigenvecs = left_eigenvecs[:, mask]343  num_stationary_eigenvecs = np.shape(left_eigenvecs)[1]344  if num_stationary_eigenvecs != 1:345    raise ValueError('Expected 1 stationary distribution, but found %d' %346                     num_stationary_eigenvecs)347  left_eigenvecs *= 1. / sum(left_eigenvecs)348  return left_eigenvecs.real.flatten()349def print_results(payoff_tables,350                  payoffs_are_hpt_format,351                  rhos=None,352                  rho_m=None,353                  c=None,354                  pi=None):355  """Prints the finite-population analysis results."""356  print('Payoff tables:\n')357  if payoffs_are_hpt_format:358    for payoff_table in payoff_tables:359      print(payoff_table())360  else:361    print(payoff_tables)362  if rho_m is not None:363    print('\nNeutral fixation probability (rho_m):\n', rho_m)364  if rhos is not None and rho_m is not None:365    print('\nFixation probability matrix (rho_{r,s}/rho_m):\n',366          np.around(rhos / rho_m, decimals=2))367  if c is not None:368    print('\nMarkov transition matrix (c):\n', np.around(c, decimals=2))369  if pi is not None:370    print('\nStationary distribution (pi):\n', pi)371def sweep_pi_vs_epsilon(payoff_tables,372                        strat_labels=None,373                        warm_start_epsilon=None,374                        visualize=False,375                        return_epsilon=False,376                        min_iters=10,377                        max_iters=100,378                        min_epsilon=1e-14,379                        num_strats_to_label=10,380                        legend_sort_clusters=False):381  """Computes infinite-alpha distribution for a range of perturbations.382  The range of response graph perturbations is defined in epsilon_list.383  Note that min_iters and max_iters is necessary as it may sometimes appear the384  stationary distribution has converged for a game in the first few iterations,385  where in reality a sufficiently smaller epsilon is needed for the distribution386  to first diverge, then reconverge. This behavior is dependent on both the387  payoff structure and bounds, so the parameters min_iters and max_iters can be388  used to fine-tune this.389  Args:390    payoff_tables: List of game payoff tables, one for each agent identity.391      Each payoff_table may be either a numpy array, or a392      _PayoffTableInterface object.393    strat_labels: Human-readable strategy labels. See get_strat_profile_labels()394      in utils.py for formatting details.395    warm_start_epsilon: Initial value of epsilon to use.396    visualize: Plot the sweep results.397    return_epsilon: Whether to return the final epsilon used.398    min_iters: the minimum number of sweep iterations.399    max_iters: the maximum number of sweep iterations.400    min_epsilon: the minimum value of epsilon to be tested, at which point the401      sweep terminates (if not converged already).402    num_strats_to_label: Number of strats to label in legend403    legend_sort_clusters: If true, strategies in the same cluster are sorted in404      the legend according to orderings for earlier alpha values. Primarily for405      visualization purposes! Rankings for lower alpha values should be406      interpreted carefully.407  Returns:408   pi: AlphaRank stationary distribution.409   epsilon: The AlphaRank transition matrix noise level resulting from sweep.410  """411  payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)412  num_populations = len(payoff_tables)413  num_strats_per_population = utils.get_num_strats_per_population(414      payoff_tables, payoffs_are_hpt_format)415  if num_populations == 1:416    num_profiles = num_strats_per_population[0]417  else:418    num_profiles = utils.get_num_profiles(num_strats_per_population)419  assert (strat_labels is None or isinstance(strat_labels, dict)420          or (len(strat_labels) == num_profiles))421  pi_list = np.empty((num_profiles, 0))422  pi, alpha, m = None, None, None  # Unused in infinite-alpha regime423  epsilon_list = []424  epsilon_pi_hist = {}425  num_iters = 0426  epsilon_mult_factor = 0.5427  alpharank_succeeded_once = False428  if warm_start_epsilon is not None:429    epsilon = warm_start_epsilon430  else:431    epsilon = 0.5432  while True:433    try:434      pi_prev = pi435      _, _, pi, _, _ = compute(payoff_tables, m=m, alpha=alpha,436                               use_inf_alpha=True, inf_alpha_eps=epsilon)437      epsilon_pi_hist[epsilon] = pi438      # Stop when pi converges439      if num_iters > min_iters and np.allclose(pi, pi_prev):440        break441      epsilon *= epsilon_mult_factor442      num_iters += 1443      alpharank_succeeded_once = True444      assert num_iters < max_iters, ('Alpharank stationary distr. not found'445                                     'after {} iterations of pi_vs_epsilon'446                                     'sweep'.format(num_iters))447    except ValueError as _:448      print('Error: ', _, epsilon, min_epsilon)449      # Case where epsilon has been decreased beyond desirable limits but no450      # distribution found.451      assert epsilon >= min_epsilon, ('AlphaRank stationary distr. not found &'452                                      'epsilon < min_epsilon.')453      # Case where epsilon >= min_epsilon, but still small enough that it causes454      # causes exceptions due to precision issues. So increase it.455      epsilon /= epsilon_mult_factor456      # Case where alpharank_succeeded_once (i.e., epsilon_list and pi_list have457      # at least one entry), and a) has not converged yet and b) failed on this458      # instance due to epsilon being too small. I.e., the rate of decreasing459      # of epsilon is too high.460      if alpharank_succeeded_once:461        epsilon_mult_factor = (epsilon_mult_factor+1.)/2.462        epsilon *= epsilon_mult_factor463  epsilon_list, pi_list = zip(*[(epsilon, epsilon_pi_hist[epsilon])464                                for epsilon in sorted(epsilon_pi_hist.keys(),465                                                      reverse=True)])466  pi_list = np.asarray(pi_list)467  if visualize:468    if strat_labels is None:469      strat_labels = utils.get_strat_profile_labels(payoff_tables,470                                                    payoffs_are_hpt_format)471    alpharank_visualizer.plot_pi_vs_alpha(472        pi_list.T,473        epsilon_list,474        num_populations,475        num_strats_per_population,476        strat_labels,477        num_strats_to_label=num_strats_to_label,478        legend_sort_clusters=legend_sort_clusters,479        xlabel=r'Infinite-AlphaRank Noise $\epsilon$')480  if return_epsilon:481    return pi_list[-1], epsilon_list[-1]482  else:483    return pi_list[-1]484def sweep_pi_vs_alpha(payoff_tables,485                      strat_labels=None,486                      warm_start_alpha=None,487                      visualize=False,488                      return_alpha=False,489                      m=50,490                      rtol=1e-5,491                      atol=1e-8,492                      num_strats_to_label=10,493                      legend_sort_clusters=False):494  """Computes stationary distribution, pi, for range of selection intensities.495  The range of selection intensities is defined in alpha_list and corresponds496  to the temperature of the Fermi selection function.497  Args:498    payoff_tables: List of game payoff tables, one for each agent identity. Each499      payoff_table may be either a numpy array, or a _PayoffTableInterface500      object.501    strat_labels: Human-readable strategy labels. See get_strat_profile_labels()502      in utils.py for formatting details.503    warm_start_alpha: Initial value of alpha to use.504    visualize: Plot the sweep results.505    return_alpha: Whether to return the final alpha used.506    m: AlphaRank population size.507    rtol: The relative tolerance parameter for np.allclose calls.508    atol: The absolute tolerance parameter for np.allclose calls.509    num_strats_to_label: Number of strats to label in legend510    legend_sort_clusters: If true, strategies in the same cluster are sorted in511      the legend according to orderings for earlier alpha values. Primarily for512      visualization purposes! Rankings for lower alpha values should be513      interpreted carefully.514  Returns:515   pi: AlphaRank stationary distribution.516   alpha: The AlphaRank selection-intensity level resulting from sweep.517  """518  payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)519  num_populations = len(payoff_tables)520  num_strats_per_population = utils.get_num_strats_per_population(521      payoff_tables, payoffs_are_hpt_format)522  if num_populations == 1:523    num_profiles = num_strats_per_population[0]524  else:525    num_profiles = utils.get_num_profiles(num_strats_per_population)526  assert (strat_labels is None or isinstance(strat_labels, dict)527          or (len(strat_labels) == num_profiles))528  pi_list = np.empty((num_profiles, 0))529  alpha_list = []530  num_iters = 0531  alpha_mult_factor = 2.532  if warm_start_alpha is not None:533    alpha = warm_start_alpha534    alpharank_succeeded_once = False535  else:536    alpha = 1e-4  # Reasonable default for most games, can be user-overridden537  while 1:538    try:539      _, _, pi, _, _ = compute(payoff_tables, alpha=alpha, m=m)540      pi_list = np.append(pi_list, np.reshape(pi, (-1, 1)), axis=1)541      alpha_list.append(alpha)542      # Stop when pi converges543      if num_iters > 0 and np.allclose(pi, pi_list[:, num_iters - 1], rtol,544                                       atol):545        break546      alpha *= alpha_mult_factor547      num_iters += 1548      alpharank_succeeded_once = True549    except ValueError as _:550      if warm_start_alpha is not None and not alpharank_succeeded_once:551        # When warm_start_alpha is used, there's a chance that552        # the initial warm_start_alpha is too large and causes exceptions due to553        # the Markov transition matrix being reducible. So keep decreasing until554        # a single success occurs.555        alpha /= 2556      elif not np.allclose(pi_list[:, -1], pi_list[:, -2], rtol, atol):557        # Sweep stopped due to multiple stationary distributions, but pi had558        # not converged due to the alpha scaling being too large.559        alpha /= alpha_mult_factor560        alpha_mult_factor = (alpha_mult_factor + 1.) / 2.561        alpha *= alpha_mult_factor562      else:563        break564  if visualize:565    if strat_labels is None:566      strat_labels = utils.get_strat_profile_labels(payoff_tables,567                                                    payoffs_are_hpt_format)568    alpharank_visualizer.plot_pi_vs_alpha(569        pi_list.T,570        alpha_list,571        num_populations,572        num_strats_per_population,573        strat_labels,574        num_strats_to_label=num_strats_to_label,575        legend_sort_clusters=legend_sort_clusters)576  if return_alpha:577    return pi, alpha578  else:579    return pi580def compute_and_report_alpharank(payoff_tables,581                                 m=50,582                                 alpha=100,583                                 verbose=False,584                                 num_top_strats_to_print=8):585  """Computes and visualizes Alpha-Rank outputs.586  Args:587    payoff_tables: List of game payoff tables, one for each agent identity. Each588      payoff_table may be either a numpy array, or a _PayoffTableInterface589      object.590    m: Finite population size.591    alpha: Fermi distribution temperature parameter.592    verbose: Set to True to print intermediate results.593    num_top_strats_to_print: Number of top strategies to print.594  Returns:595    pi: AlphaRank stationary distribution/rankings.596  """597  payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)598  rhos, rho_m, pi, _, _ = compute(payoff_tables, m=m, alpha=alpha)599  strat_labels = utils.get_strat_profile_labels(payoff_tables,600                                                payoffs_are_hpt_format)601  if verbose:602    print_results(payoff_tables, payoffs_are_hpt_format, pi=pi)603  utils.print_rankings_table(604      payoff_tables,605      pi,606      strat_labels,607      num_top_strats_to_print=num_top_strats_to_print)608  m_network_plotter = alpharank_visualizer.NetworkPlot(609      payoff_tables, rhos, rho_m, pi, strat_labels, num_top_profiles=8)610  m_network_plotter.compute_and_draw_network()611  return pi612def compute(payoff_tables,613            m=50,614            alpha=100,615            use_local_selection_model=True,616            verbose=False,617            use_inf_alpha=False,618            inf_alpha_eps=0.01):619  """Computes the finite population stationary statistics.620  Args:621    payoff_tables: List of game payoff tables, one for each agent identity. Each622      payoff_table may be either a numpy array, or a _PayoffTableInterface623      object.624    m: Finite population size.625    alpha: Fermi distribution temperature parameter.626    use_local_selection_model: Enable local evolutionary selection model, which627      considers fitness against the current opponent only, rather than the628      global population state.629    verbose: Set to True to print intermediate results.630    use_inf_alpha: Use infinite-alpha alpharank model.631    inf_alpha_eps: Noise term to use in infinite-alpha alpharank model.632  Returns:633    rhos: Matrix of strategy-to-strategy fixation probabilities.634    rho_m: Neutral fixation probability.635    pi: Finite population stationary distribution.636    num_strats: Number of available strategies.637  """638  payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)639  num_populations = len(payoff_tables)640  num_strats_per_population = utils.get_num_strats_per_population(641      payoff_tables, payoffs_are_hpt_format)642  # Handles the trivial case of Markov chain with one state643  if np.array_equal(num_strats_per_population,644                    np.ones(len(num_strats_per_population))):645    rhos = np.asarray([[1]])646    rho_m = 1. / m if not use_inf_alpha else 1647    num_profiles = 1648    pi = np.asarray([1.])649    return rhos, rho_m, pi, num_profiles, num_strats_per_population650  if verbose:651    print('Constructing c matrix')652    print('num_strats_per_population:', num_strats_per_population)653  if num_populations == 1:654    # User fast closed-form analysis for constant-sum single-population games655    game_is_constant_sum, payoff_sum = utils.check_is_constant_sum(656        payoff_tables[0], payoffs_are_hpt_format)657    if verbose:658      print('game_is_constant_sum:', game_is_constant_sum, 'payoff sum: ',659            payoff_sum)660    # Single-population/symmetric game just uses the first player's payoffs661    c, rhos = _get_singlepop_transition_matrix(662        payoff_tables[0],663        payoffs_are_hpt_format,664        m,665        alpha,666        game_is_constant_sum,667        use_local_selection_model,668        payoff_sum,669        use_inf_alpha=use_inf_alpha,670        inf_alpha_eps=inf_alpha_eps)671    num_profiles = num_strats_per_population[0]672  else:673    c, rhos = _get_multipop_transition_matrix(674        payoff_tables,675        payoffs_are_hpt_format,676        m,677        alpha,678        use_inf_alpha=use_inf_alpha,679        inf_alpha_eps=inf_alpha_eps)680    num_profiles = utils.get_num_profiles(num_strats_per_population)681  pi = _get_stationary_distr(c)682  rho_m = 1. / m if not use_inf_alpha else 1  # Neutral fixation probability683  if verbose:684    print_results(payoff_tables, payoffs_are_hpt_format, rhos, rho_m, c, pi)685  return rhos, rho_m, pi, num_profiles, num_strats_per_population686def suggest_alpha(payoff_tables, tol=.1):687  """Suggests an alpha for use in alpha-rank.688  The suggested alpha is approximately the smallest possible alpha such that689  the ranking has 'settled out'. It is calculated as690  -ln(tol)/min_gap_between_payoffs.691  The logic behind this settling out is that the fixation probabilities can be692  expanded as a series, and the relative size of each term in this series693  changes with alpha. As alpha gets larger and larger, one of the terms in694  this series comes to dominate, and this causes the ranking to settle695  down. Just how fast this domination happens is easy to calculate, and this696  function uses it to estimate the alpha by which the ranking has settled.697  You can find further discussion at the PR:698  https://github.com/deepmind/open_spiel/pull/403699  Args:700    payoff_tables: List of game payoff tables, one for each agent identity. Each701      payoff_table may be either a numpy array, or a _PayoffTableInterface702      object.703    tol: the desired gap between the first and second terms in the fixation704      probability expansion. A smaller tolerance leads to a larger alpha, and705      a 'more settled out' ranking.706  Returns:707    A suggested alpha.708  """709  payoffs_are_hpt_format = utils.check_payoffs_are_hpt(payoff_tables)710  num_strats_per_population = utils.get_num_strats_per_population(711      payoff_tables, payoffs_are_hpt_format)712  num_profiles = utils.get_num_profiles(num_strats_per_population)713  gap = np.inf714  for id_row_profile in range(num_profiles):715    row_profile = utils.get_strat_profile_from_id(num_strats_per_population,716                                                  id_row_profile)717    next_profile_gen = utils.get_valid_next_profiles(num_strats_per_population,718                                                     row_profile)719    for index_population_that_changed, col_profile in next_profile_gen:720      payoff_table_k = payoff_tables[index_population_that_changed]721      f_r = _get_payoff(payoff_table_k, payoffs_are_hpt_format, col_profile,722                        index_population_that_changed)723      f_s = _get_payoff(payoff_table_k, payoffs_are_hpt_format, row_profile,724                        index_population_that_changed)725      if f_r > f_s:726        gap = min(gap, f_r - f_s)...

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